diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzkqeg" "b/data_all_eng_slimpj/shuffled/split2/finalzzkqeg" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzkqeg" @@ -0,0 +1,5 @@ +{"text":"\\section{INTRODUCTION}\n\nThe discrete Fourier transform (DFT) can be applied in error correcting codes and code-based cryptography. \nThe cyclotomic DFT method \\cite{Fedorenko03,Fedorenko04} is the best one for computing the DFT over finite field. \nAlexey Maevskiy pointed that formula \\hbox{\\cite[(6)]{Fedorenko04}} in the example of paper \\cite{Fedorenko04} \nhad not been proved. \nWe have corrected this mistake and introduced the proof.\n\n\\section{BASIC NOTIONS AND DEFINITIONS}\n\nThe DFT of length ${n \\mid 2^m-1}$ of a vector \n$f = (f_i)$, \\, $i \\in [0,n-1]$, $f_i \\in GF(2^m)$,\nis the vector $F = (F_j)$\n$$F_j = \\sum_{i=0}^{n-1} f_i \\alpha^{ij}, \\quad j \\in [0,n-1],$$ \nwhere $\\alpha$ is an element of order $n$ in $GF(2^m)$. \nLet us write the DFT in matrix form\n\n\\begin{equation}\nF = Wf,\n\\end{equation}\nwhere $W = (\\alpha^{ij})$, \\, $i,j \\in [0,n-1]$, is a Vandermonde matrix.\nWe assume that the length of the $n$-point Fourier transform\nover $GF(2^m)$ is $n = 2^m - 1$. \n\nLet us consider cyclotomic cosets modulo\n$n=2^m-1$ over $GF(2)$ \n\n\\begin{eqnarray*}\n\\{c_0\\} = \\{0\\}\\\\\n \\{c_1,c_1 2,c_1 2^2, \\ldots,c_1 2^{m_1-1}\\},\\\\\n\\ldots,\\\\\n \\{c_l,c_l 2,c_l 2^2, \\ldots,c_l 2^{m_l-1}\\},\n\\end{eqnarray*}\nwhere $c_k \\equiv c_k 2^{m_k} \\bmod n$,\n$l+1$ is the number of cyclotomic cosets modulo $n$ over $GF(2)$.\n\nLet us introduce the set of indices modulo $n$\n\n$$\nZ = (Z_i) = (c_0,c_1,c_12,c_12^2, \\ldots,c_12^{m_1-1}, \\ldots,$$\n$$c_l,c_l2,c_l2^2, \\ldots,c_l2^{m_l-1}), \\quad i \\in [0,n-1].\n$$\n\nThen, we define a permutation matrix $\\Pi=(\\Pi_{i,j})$, \\, $i,j \\in [0,n-1]$,\n\n$$ \n\\Pi_{i,j} = \n\\left\\{\n\\begin{matrix}\n1, & \\hbox{ if } j = Z_i, \\quad i \\in [0,n-1] \\cr\n0, & \\hbox{ otherwise}. \\cr\n\\end{matrix}\n\\right.\n$$\n\nLet us denote a basis\n$\\beta_k=\\left(\\beta_{k,0}, \\ldots ,\\beta_{k,m_k-1}\\right)$ of the subfield $GF(2^{m_k}) \\subset GF(2^m)$.\n\nThen we can write the cyclotomic DFT \\cite{Fedorenko03,Fedorenko04}\n\n\\begin{equation*}\nF_j=\\sum_{k=0}^l\\sum_{s=0}^{m_k-1}a_{k,j,s}\\left(\\sum_{p=0}^{m_k-1}\n\\beta_{k,s}^{2^p}f_{c_k 2^p}\\right),\n\\end{equation*}\nwhere $a_{k,j,s}\\in GF(2)$. \n\nThis equation can be represented in matrix form as\n\n\\begin{equation}\nF=AL (\\Pi f),\n\\end{equation} \nwhere $A$ is a matrix with elements $a_{k,j,s}\\in GF(2)$ \nand $L$ is a block diagonal matrix with elements $\\beta_{k,s}^{2^p}$.\nIf one chooses the normal basis $\\beta_k$, then all the blocks of \nthe matrix $L$ are circulant matrices. \n\nThe inverse DFT in the field $GF(2^m)$ is \n$$\nf=W^{-1}F.\n$$\n\nIt is easily shown that \n\\begin{equation}\nW^{-1} = E W,\n\\end{equation} \nwhere $E$ is a matrix\n\n$$ \nE =\n\\begin{pmatrix}\n1 & 0 & 0 & \\cdot & 0 & \\cdot & 0 & 0 \\cr\n0 & 0 & 0 & \\cdot & 0 & \\cdot & 0 & 1 \\cr\n0 & 0 & 0 & \\cdot & 0 & \\cdot & 1 & 0 \\cr\n\\cdot & \\cdot &\\cdot & \\cdot & \\cdot& \\cdot & \\cdot& \\cdot \\cr\n0 & 0 & 0 & \\cdot & 1 & \\cdot & 0 & 0 \\cr\n\\cdot & \\cdot &\\cdot & \\cdot & \\cdot& \\cdot & \\cdot& \\cdot \\cr\n0 & 0 & 1 & \\cdot & 0 & \\cdot & 0 & 0 \\cr\n0 & 1 & 0 & \\cdot & 0 & \\cdot & 0 & 0 \\cr\n\\end{pmatrix}.\n$$\n\n\\bigskip\n\n\\section{THE INVERSE CYCLOTOMIC DFT}\n\nSince both matrices $A$ and $L$ are invertible, from (2) \nthe following representation of the inverse DFT can be derived\n\\begin{equation*}\n\\Pi f = L^{-1} A^{-1} F.\n\\end{equation*}\n\n\\begin{Lemma}[\\cite{Hong}]\n\\label{lemma1}\nSuppose $\\beta_k$ are the normal bases,\nthen it is possible to show that blocks of $L^{-1}$ consist\nof elements of bases $\\beta_{k}'$ which are dual to $\\beta_{k}$,\nthat is, the blocks of $L^{-1}$ are also circulant matrices. \n\\end{Lemma}\n\n\\begin{Theorem}\n\\label{theorem1}\nThe inverse cyclotomic DFT for $GF(2^m)$ is\n$$\n(\\Pi E) F = L^{-1} A^{-1} f.\n$$\n\\end{Theorem}\n\n\\begin{IEEEproof}\nFrom (1) and (2) we have \n\n$$\nW = AL\\Pi\n$$ \nand \n$$\nW^{-1} = \\Pi^{-1} L^{-1} A^{-1}.\n$$\n\nFrom the last formula and (3) we obtain \n$$\nW^{-1} = E W = \\Pi^{-1} L^{-1} A^{-1} \n$$\nand\n$$\nW = E^{-1} \\Pi^{-1} L^{-1} A^{-1}. \n$$\n\nHence,\n$$\nF = W f = (E^{-1} \\Pi^{-1}) L^{-1} A^{-1} f.\n$$\nand\n$$\n(\\Pi E) F = L^{-1} A^{-1} f.\n$$\n\\end{IEEEproof}\n\n\\section{EXAMPLES}\n\n\\subsection{DFT of length $n=7$}\n\nLet $(\\gamma,\\gamma^2,\\gamma^4)$ be a normal basis of $GF(2^3)$, \nwhere $\\gamma=\\alpha^3$ and $\\alpha$ is a root of the primitive polynomial $x^3+x+1$. \nThen the cyclotomic DFT can be represented as\n\n$$\nF =\n\\begin{pmatrix}\nF_0\\\\\nF_1\\\\\nF_2\\\\\nF_3\\\\\nF_4\\\\\nF_5\\\\\nF_6\\\\\n\\end{pmatrix}=\n\\begin{pmatrix}\n1&1&1&1&1&1&1\\\\\n1&0&1&1&1&0&0\\\\\n1&1&0&1&0&1&0\\\\\n1&1&0&0&1&0&1\\\\\n1&1&1&0&0&0&1\\\\\n1&0&0&1&0&1&1\\\\\n1&0&1&0&1&1&0\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\n1&0 &0 &0 &0 &0 &0 \\\\\n0&\\gamma^1&\\gamma^2&\\gamma^4 &0 &0 &0 \\\\\n0&\\gamma^2&\\gamma^4&\\gamma^1 &0 &0 &0 \\\\\n0&\\gamma^4&\\gamma^1&\\gamma^2 &0 &0 &0 \\\\\n0&0 &0 &0 &\\gamma^1&\\gamma^2&\\gamma^4\\\\\n0&0 &0 &0 &\\gamma^2&\\gamma^4&\\gamma^1\\\\\n0&0 &0 &0 &\\gamma^4&\\gamma^1&\\gamma^2\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\n 1 &0 &0 &0 &0 &0 &0 \\\\\n 0 &1 &0 &0 &0 &0 &0 \\\\\n 0 &0 &1 &0 &0 &0 &0 \\\\\n 0 &0 &0 &0 &1 &0 &0 \\\\\n 0 &0 &0 &1 &0 &0 &0 \\\\\n 0 &0 &0 &0 &0 &0 &1 \\\\\n 0 &0 &0 &0 &0 &1 &0 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\nf_0\\\\\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\\\\\nf_5\\\\\nf_6\\\\\n\\end{pmatrix}\n= AL \\Pi f.\n$$\n\nUsing Theorem~\\ref{theorem1}, we obtain the inverse cyclotomic DFT \n\n$$\n(\\Pi E) F\n$$\n$$ =\n\\begin{pmatrix}\n 1 &0 &0 &0 &0 &0 &0 \\\\\n 0 &1 &0 &0 &0 &0 &0 \\\\\n 0 &0 &1 &0 &0 &0 &0 \\\\\n 0 &0 &0 &0 &1 &0 &0 \\\\\n 0 &0 &0 &1 &0 &0 &0 \\\\\n 0 &0 &0 &0 &0 &0 &1 \\\\\n 0 &0 &0 &0 &0 &1 &0 \\\\\n\\end{pmatrix}\n\\begin{pmatrix}\n 1 &0 &0 &0 &0 &0 &0 \\\\\n 0 &0 &0 &0 &0 &0 &1 \\\\\n 0 &0 &0 &0 &0 &1 &0 \\\\\n 0 &0 &0 &0 &1 &0 &0 \\\\\n 0 &0 &0 &1 &0 &0 &0 \\\\\n 0 &0 &1 &0 &0 &0 &0 \\\\\n 0 &1 &0 &0 &0 &0 &0 \\\\\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\nF_0\\\\\nF_1\\\\\nF_2\\\\\nF_3\\\\\nF_4\\\\\nF_5\\\\\nF_6\\\\\n\\end{pmatrix}\n=\n\\begin{pmatrix}\n1&0 &0 &0 &0 &0 &0 \\\\\n0&\\gamma^1&\\gamma^2&\\gamma^4 &0 &0 &0 \\\\\n0&\\gamma^2&\\gamma^4&\\gamma^1 &0 &0 &0 \\\\\n0&\\gamma^4&\\gamma^1&\\gamma^2 &0 &0 &0 \\\\\n0&0 &0 &0 &\\gamma^1&\\gamma^2&\\gamma^4\\\\\n0&0 &0 &0 &\\gamma^2&\\gamma^4&\\gamma^1\\\\\n0&0 &0 &0 &\\gamma^4&\\gamma^1&\\gamma^2\n\\end{pmatrix}\n$$\n$$\n\\times\n\\begin{pmatrix}\n1&1&1&1&1&1&1\\\\\n1&0&0&1&1&1&0\\\\\n1&1&0&1&0&0&1\\\\\n1&0&1&0&0&1&1\\\\\n1&1&0&0&1&0&1\\\\\n1&1&1&0&0&1&0\\\\\n1&0&1&1&1&0&0\n\\end{pmatrix}\n\\begin{pmatrix}\nf_0\\\\\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\\\\\nf_5\\\\\nf_6\\\\\n\\end{pmatrix}\n= L^{-1} A^{-1} f.\n$$\n\nFinally note that the last formula coincides with formula \\hbox{\\cite[(6)]{Fedorenko04}}.\n\n$$\n\\begin{pmatrix}\nF_0\\\\\nF_6\\\\\nF_5\\\\\nF_3\\\\\nF_4\\\\\nF_1\\\\\nF_2\\\\\n\\end{pmatrix}\n= L^{-1} A^{-1}\n\\begin{pmatrix}\nf_0\\\\\nf_1\\\\\nf_2\\\\\nf_3\\\\\nf_4\\\\\nf_5\\\\\nf_6\n\\end{pmatrix}.\n$$\n\nThis algorithm requires 6 multiplications and \n24 additions and appears to be the best known 7-point DFT for $GF(2^3)$.\n\n\\subsection{DFT of length $n=15$}\n\nLet $(\\alpha^{ 3},\\alpha^{ 6},\\alpha^{12},\\alpha^{ 9})$ and $(\\alpha^{11},\\alpha^{ 7},\\alpha^{14},\\alpha^{13})$\nbe the normal bases of $GF(2^3)$, \nwhere $\\alpha$ is a root of the primitive polynomial $x^4+x+1$. \nThen the cyclotomic DFT can be represented as formula (4) or $F=AL (\\Pi f)$.\nThe inverse cyclotomic DFT can be written as formula (5) or $(\\Pi E) F = L^{-1} A^{-1} f$.\n\n\\section*{ACKNOWLEDGMENT}\n\nThe author would like to thank Alexey Maevskiy for pointing out a methodological mistake in the paper \\cite{Fedorenko04},\nand Peter Trifonov for helpful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this paper we shall use a manifestly covariant form of statistical\nmechanics which has more general structure than the standard forms\nof relativistic statistical mechanics, but which reduces to those theories\nin a certain limit, to be described precisely below. These theories,\nwhich are characterized classically by mass-shell constraints, and the\nuse, in quantum field theory, of fields which are constructed on the\nbasis of on-mass-shell free fields, are associated with the\nstatistical treatment of {\\it world lines} and hence, considerable\ncoherence (in terms of the macroscopic structure of whole world lines\nas the elementary objects of the theory) is implied. In\nnonrelativistic statistical mechanics, the elementary objects of the\ntheory are points.\n The relativistic analog of this essentially structureless foundation for a statistical theory is the set of points in spacetime, i.e., the so-called {\\it events}, not the world lines (Currie, Jordan and Sudarshan \\cite {Sudarshan} have discussed the difficulty of constructing a relativistic mechanics on the basis of world lines).\n\nThe mass of particles in a mechanical theory of events is necessarily a dynamical variable, since the classical phase space of the relativistic set of events consists of the spacetime and energy-momentum coordinates $\\{{\\bf q}_i, t_i;\\ {\\bf p}_i,E_i\\}$, with no {\\it a priori} constraint on the relation between the ${\\bf p}_i$ and the $E_i$, and hence such theories are ``off-shell''. It is well known from the work of Newton and Wigner \\cite{NW} that on-shell relativistic quantum theories such as those governed by Klein-Gordon or Dirac type equations do not provide local descriptions (the wave functions corresponding to localized particles are spread out); for such theories the notion of ensembles over local initial conditions is difficult to formulate. The off-shell theory that\nwe shall use here is, however, precisely local in both its first and\nsecond quantized forms \\cite{AH,Shnerb}.\n\nWe finally remark that the standard formulations of quantum relativistic\nstatistical mechanics, and quantum field theory at finite temperature, lack\nmanifest covariance on a fundamental level. As for nonrelativistic statistical\nmechanics, the partition function is described by the Hamiltonian, which is not\nan invariant object, and hence thermodynamic mean values do not have tensor\nproperties. [One could consider the invariant $p_\\mu n^\\mu$ in place of the\nHamiltonian \\cite{Synge}, where $n^\\mu$ is a unit four-vector; this construction (supplemented by a spacelike vector othogonal to $n^\\mu$) implies an induced\nrepresentation for spacetime. The quantity that takes the place of the\nparameter $t$ is then $x_\\mu n^\\mu$; in the corresponding quantum mechanics,\nthe space parts of (induced form of) the momentum do not commute with\nthis time variable. Some of the problems associated with this construction are closely related to those pointed out by Currie, Jordan and Sudarshan \\cite{Sudarshan}, for which\ndifferent world lines are predicted dynamically by the change in the form of the effective Hamiltonian in different frames.] Since the form of such a theory is not constrained by covariance requirements, its dynamical structure\nand predictions may be different than for a theory which satisifies these\nrequirements. For example, the canonical distribution of Pauli \\cite{Pauli}\nfor the free Boltzmann gas has a high temperature limit in which the energy is\ngiven by $3k_BT$, which does not correspond to any known\nequipartition rule, but for the corresponding distribution for the\nmanifestly covariant theory, the limit is $2k_BT$, corresponding to\n${1\\over 2}k_BT$ for each of the four relativistic degrees of freedom\n\\cite{HSP,BH1}. For the quantum field theories at finite temperature,\nthe path integral formulation \\cite {Kap1} replaces the Hamiltonian in\nthe canonical exponent by the Lagrangian due to the infinite product of factors $\\langle\\phi \\vert \\pi\\rangle$ (transition matrix element of the\ncanonical field and its conjugate required to give a Weyl ordered Hamiltonian\nits numerical value). However, it is the $t$ variable which is analytically\ncontinued to construct the finite temperature canonical ensemble, completely\nremoving the covariance of the theoretical framework. One may argue that some\nframe has to be chosen for the statistical theory to be developed, and perhaps\neven for temperature to have a meaning, but as we have remarked above, the\nrequirement of relativistic covariance has dynamical consequences (note that the model Lagrangians used in the non-covariant formulations are established with\nthe criterion of relativistic covariance in mind), and we argue that the choice\nof a frame, if necessary for some physical reason, such as the definition and\nmeasurement of temperature, should be made in the framework of a manifestly\ncovariant structure.\n\nIn the framework of a manifestly covariant relativistic statistical\nmechanics, the dynamical evolution of a system of $N$ particles, for the\nclassical case, is governed by equations of motion that are of the form\nof Hamilton equations for the motion of $N$ $events$ which generate the\nspace-time trajectories (particle world lines) as functions of a\ncontinuous Poincar\\'{e}-invariant parameter $\\tau $ \\cite{Stu,HP}, usually \nreferred to as a ``proper time''. These events are characterized by\ntheir positions $q^\\mu =(t,{\\bf q})$ and energy-momenta $p^\\mu =(\nE,{\\bf p})$ in an $8N$-dimensional phase-space. For the quantum case, the\nsystem is characterized by the wave function $\\psi _\\tau (q_1,q_2,\\ldots\n,q_N)\\in L ^2(R^{4N}),$ with the measure $d^4q_1d^4q_2\\cdots d^4q_N\\equiv\nd^{4N}q,$ $(q_i\\equiv q_i^\\mu ;\\;\\;\\mu =0,1,2,3;\\;\\;i=1,2,\\ldots ,N),$\ndescribing the distribution of events, which evolves with a generalized\nSchr\\\"{o}dinger equation \\cite{HP}. The collection of events (called\n``concatenation'' \\cite{AHL}) along each world line corresponds to a\n{\\it particle,} and hence, the evolution of the state of the $N$-event\nsystem describes, {\\it a posteriori,} the history in space and time of\nan $N$-particle system.\n\nFor a system of $N$ interacting events (and hence, particles) one takes\n\\cite{HP}\n\\beq\nK=\\sum _i\\frac{p_i^\\mu p_{i\\mu }}{2M}+V(q_1,q_2,\\ldots ,q_N),\n\\eeq\nwhere $M$ is a given fixed parameter (an intrinsic property of the\nparticles), with the dimension of mass, taken to be the same for all the\nparticles of the system. The Hamilton equations are\n$$\\frac{dq_i^\\mu }{d\\tau }=\\frac{\\partial K}{\\partial p_{i\\mu }}=\\frac{p_\ni^\\mu }{M},$$\n\\beq\n\\frac{dp_i^\\mu }{d\\tau }=-\\frac{\\partial K}{\\partial q_{i\\mu }}=-\\frac{\n\\partial V}{\\partial q_{i\\mu }}.\n\\eeq\nIn the quantum theory, the generalized Schr\\\"{o}dinger equation\n\\beq\ni\\frac{\\partial }{\\partial \\tau }\\psi _\\tau (q_1,q_2,\\ldots ,q_N)=K\n\\psi _\\tau (q_1,q_2,\\ldots ,q_N)\n\\eeq\ndescribes the evolution of the $N$-body wave function\n$\\psi _\\tau (q_1,q_2,\\ldots ,q_N).$ To illustrate the meaning of this\nwave function, consider the case of a single free event. In this case\n(1.3) has the formal solution\n\\beq\n\\psi _\\tau (q)=(e^{-iK_0\\tau }\\psi _0)(q)\n\\eeq\nfor the evolution of the free wave packet. Let us represent $\\psi _\\tau\n(q)$ by its Fourier transform, in energy-momentum space:\n\\beq\n\\psi _\\tau (q)=\\frac{1}{(2\\pi )^2}\\int d^4pe^{-i\\frac{p^2}{2M}\\tau }\ne^{ip\\cdot q}\\psi _0(p),\n\\eeq\nwhere $p^2\\equiv p^\\mu p_\\mu ,$ $p\\cdot q\\equiv p^\\mu q_\\mu ,$ and $\\psi\n_0(p)$ corresponds to the initial state. Applying the Ehrenfest arguments\nof stationary phase to obtain the principal contribution to $\\psi _\\tau\n(q)$ for a wave packet at $p_c^\\mu ,$ one finds ($p_c^\\mu $ is the peak\nvalue in the distribution $\\psi _0(p))$\n\\beq\nq_c^\\mu \\simeq \\frac{p_c^\\mu }{M}\\tau ,\n\\eeq\nconsistent with the classical equations (1.2). Therefore,\nthe central peak of the wave packet moves along the classical\ntrajectory of an event, i.e., the classical world line.\n\nIt is clear from the form of (1.3) that one can construct relativistic\ntransport theory in a form analogous to that of the nonrelativistic theory; a\nrelativistic Boltzmann equation and its consequences, for example, was studied\nin ref. \\cite{HSS}.\n\nSince, in the Hilbert space $L^2(R^4)$ the operators $x^\\mu ,p^\\nu $ obey the canonical commutation relations $(g^{\\mu \\nu }={\\rm diag}(-,+,+,+))$\n\\beq\n[x^\\mu ,p^\\nu ]=i\\hbar g^{\\mu \\nu },\n\\eeq\nthe uncertainty relations\n\\beq\n\\triangle x\\;\\triangle p\\geq\\frac{\\hbar }{2},\\;\\;\\;\n\\triangle t\\;\\triangle E\\geq\\frac{\\hbar }{2}\n\\eeq\nfollow directly from the mathematical structure of the theory, and are\non the same footing (with the usual statistical interpretation\n\\cite{FW}). The dispertion $\\triangle t$ is a property of the wave\nfunction $\\psi _\\tau (q)$ at a given $\\tau .$\n\nArshansky and Horwitz \\cite{AH} have studied thought experiments analogous to those discussed by Landau and Peierls \\cite{LP}, within the framework of the manifestly covariant relativistic quantum theory which we are using here, and derived the (causal) Landau-Peierls relation\n\\beq\n\\triangle t\\;\\triangle p\\stackrel{>}{\\sim }\\frac{\\hbar }{c},\n\\eeq\nconcerning the time interval $\\triangle t$ during which the momentum of a particle is measured, and the momentum dispersion of the state. \n\nIn this paper we shall discuss another uncertainty relation following\ndirectly from the mathematical structure of the theory which is\nrealized on a statistical mechanical level (for an ensemble of events),\n\\beq\nT_{\\triangle V}\\;\\triangle m\\stackrel{>}{\\sim }\\frac{2\\pi \\hbar}{c^2},\n\\eeq\nwhere $\\triangle m$ is the mass dispersion around the on-shell value\ndue to the off-shellness of the events making up the ensemble, and\n$T_{\\triangle V}$ is the average passage interval in $\\tau $ for the\nevents which pass through the small (typical) four-volume $\\triangle\nV$ in the neighborhood of the $R^4$-point.\\footnote{This result is\nanalogous to the {\\it nonrelativistic} $\\triangle t\\;\\triangle E$\nrelation, which, as for (1.10), does not follow from commutation\nrelations and the Schwartz inequality. The nonrelativistic time-energy \nuncertainty relation, in fact, follows from (1.10) and (1.11) in the\n nonrelativistic \nlimit for which $\\tau \\rightarrow t,$ $\\langle E\\rangle \/M \\rightarrow 1,$\nand $\\triangle m\\;c^2 \\rightarrow \\triangle E$. This result implies\nthe existence, in the non-relativistic limit, of a residual ensemble\nover $t$, consistently with the treatment of the non-relativistic\nlimit given in ref. [7].} The four-volume $\\triangle V$ is the smallest that can be considered a macrovolume in representing the ensemble. $T_{\\triangle V}$ is related to the (average) extent of the ensemble along the time axis, through the Hamilton equation (1.2) for $\\mu =0$ (in the sense of a statistical average), \n\\beq\n\\frac{\\triangle t}{T_{\\triangle V}}=\\frac{\\langle E\\rangle }{M},\n\\eeq\nif the ensemble is constructed with the minimum time span to\ncharacterize the physical system.\n\n\\section{Ideal relativistic gas of events}\nTo describe an ideal gas of events in the grand canonical ensemble, we\nuse the expression for the number of events given in \\cite{HSP} (in the following we shall use the system of units in which $\\hbar =c=k_B=1,$ unless otherwise specified),\n\\beq\nN=\\sum _{p^\\mu }n_{p^\\mu }=\\sum _{p^\\mu }\\frac{1}{e^{(E-\\mu -\\mu _K\\frac{m^2}{2M})\/T}\\mp 1},\n\\eeq\nwhere $E\\equiv p^0,$ $m^2 \\equiv -p^2 = - p^\\mu p_\\mu ,$ and the sign in the denominator of (2.1) is determined by the event statistics in the usual way;\n$\\mu _K$ is an additional mass potential \\cite{HSP}, which arises in the\ngrand canonical ensemble as the derivative of the free energy with respect to\nthe value of the dynamical evolution function $K$, interpreted as the invariant\nmass of the system. In the kinetic theory \\cite{HSP}, $\\mu _K$ enters as a Lagrange multiplier for the equilibrium distribution for $K,$ as $\\mu $ is for $N$ and $1\/T$ for $E.$ In order to simplify subsequent considerations, we\nshall take it to be a fixed parameter. \n\nWe restrict ourselves, in the following, to the case of the events obeying Bose-Einstein statistics and use, therefore, the relation (2.1) with the minus sign in the denominator. \nTo ensure a positive-definite value for $n_{p^\\mu },$ the number density of bosons with four-momentum $p^\\mu ,$ we require that\n\\beq\nm-\\mu -\\mu _K\\frac{m^2}{2M}\\geq 0.\n\\eeq\nThe discriminant for the l.h.s. of the inequality must be nonnegative,\ni.e.,\n\\beq\n\\mu \\leq \\frac{M}{2\\mu _K}.\n\\eeq\nFor such $\\mu ,$ (2.2) has the solution\n\\beq\nm_1\\equiv \\frac{M}{\\mu _K}\\left( 1-\\sqrt{1-\\frac{2\\mu \\mu _K}{M}}\\right) \\leq\nm\\leq\n\\frac{M}{\\mu _K}\\left( 1+\\sqrt{1-\\frac{2\\mu \\mu _K}{M}}\\right) \\equiv m_2.\n\\eeq\nFor small $\\mu \\mu _K\/M,$ the region (2.4) may be approximated by\n\\beq\n\\mu \\leq m\\leq \\frac{2M}{\\mu _K}.\n\\eeq\nOne sees that $\\mu _K$ plays a fundamental role in determining an upper\nbound of the mass spectrum, in addition to the usual lower bound $m\\geq\n\\mu .$ In fact, small $\\mu _K$ admits a very large range of off-shell\nmass, and hence can be associated with the presence of strong\ninteractions \\cite{MS}. For our present purposes it will be sufficient\nto assume that the mass distribution has a finite range $m_1\\leq\nm\\leq m_2$ around the on-shell value $m_c= M\/\\mu_K$ corresponding to the \nlimiting value for which the inequality (2.3) becomes an equality.\n\nIn order to show that our results hold independent of the dimensionality of spacetime, we shall consider our ensemble in \none temporal and $D$ spatial dimensions, $D\\geq 1.$\n\nReplacing the sum over $p^\\mu $ (2.1) by an integral, $$\\sum _{p^\\mu }\\Longrightarrow \\frac{V^{(1+D)}}{(2\\pi )^{1+D}}\\int d^{1+D}p,$$\nwhere $V^{(1+D)}$ is the system's $(1+D)$-volume, and using the relation $(p^\\mu =(p^0,{\\bf p}))$ $$d^{(1+D)}p=\\frac{d^Dp}{2p^0}dm^2,\\;\\;m^2\\equiv -p^\\mu p_\\mu ,\\;\\;\\mu =0,1,\\ldots ,D,$$ one obtains for the density of events per unit $(1+D)$-volume, $n\\equiv N\/V^{(1+D)},$\n\\beq\nn=\\int _{m_1}^{m_2}\\frac{dm\\;m}{2\\pi }\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}f(p),\n\\eeq\nwith $f(p)\\equiv n_{p^\\mu },$ as given in Eq. (2.1). \nTypical average values are given by the relations\n\\beq\n\\langle p^\\mu \\rangle =\\frac{1}{n}\\int _{m_1}^{m_2}\\frac{dm\\;m}{2\\pi }\n\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}\\;p^\\mu f(p),\n\\eeq\n\\beq\n\\langle p^\\mu p^\\nu \\rangle =\\frac{1}{n}\\int _{m_1}^{m_2}\\frac{dm\\;m}{2\\pi }\n\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}\\;p^\\mu p^\\nu f(p),\\;\\;\\;{\\rm etc.}\n\\eeq\n\nTo find the expressions for the pressure and energy density in our ensemble, we study the particle energy-momentum tensor defined by the relation\\footnote{The corresponding relation of ref. \\cite{HSS} is given in four-dimensional spacetime.} \\cite{HSS}\n\\beq\nT^{\\mu \\nu }(q)=\\sum _i\\int d\\tau \\frac{p_i^\\mu p_i^\\nu }{m_c}\n\\delta ^{1+D}(q-q_i(\\tau )),\n\\eeq\nin which $m_c$ is the value around which the mass of the events making up the \nensemble is distributed. Upon integrating over a small $(1+D)$-volume \n$\\triangle V$ and taking the ensemble average, (2.9) reduces to \\cite{HSS}\n\\beq\n\\langle T^{\\mu \\nu }\\rangle =\\frac{T_{\\triangle V}}{m_c}n\\langle\np^\\mu p^\\nu \\rangle .\n\\eeq\nIn this formula, $n=N\/V$, and $T_{\\triangle V}$ is the average passage interval in\n$\\tau $ for the events which pass through $\\triangle V,$ which we discussed above. The formula (2.10) reduces, through Eq. (2.8), to\n\\beq\n\\langle T^{\\mu \\nu }\\rangle =\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^Dp^0}\\;p^\\mu p^\\nu f(p).\n\\eeq\nUsing the standard expression\n\\beq\n\\langle T^{\\mu \\nu }\\rangle =pg^{\\mu \\nu }-(p+\\rho )u^\\mu u^\\nu ,\n\\eeq\nwhere $p$ and $\\rho $ are the particle pressure and energy density,\nrespectively, we obtain $$\\rho =\\langle T^{00}\\rangle,\\;\\;\\;p=\\frac{1}{D}g^{ii}\\langle T_{ii}\\rangle ,\\;\\;i=1,2,\\ldots, D,$$ and therefore, through (2.11),\n\\beq\np=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\frac{{\\bf p}^2}{Dp^0}f(p),\n\\eeq\n\\beq\n\\rho =\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;p^0 f(p).\n\\eeq\nWe now calculate the particle number density per\nunit $D$-volume. The particle $D+1$-current is given by the\nformula \\cite{HSS}\n\\beq\nJ^\\mu (q)=\\sum _i\\int d\\tau \\frac{p^\\mu _i}{m_c}\\delta ^{1+D}(q-q_i(\\tau )),\n\\eeq\nwhich upon integrating over a small $(1+D)$-volume and taking the\naverage reduces to\n\\beq\n\\langle J^\\mu \\rangle =\\frac{T_{\\triangle V}}{m_c}n\\langle p^\\mu \\rangle ;\n\\eeq\nthen the {\\it particle} number density \\cite{Synge,Hakim} is\n\\beq\nN_0\\equiv \\langle J^0\\rangle =\\frac{T_{\\triangle V}}{m_c}n\\langle E\\rangle ,\n\\eeq\nso that, with the help of Eq. (2.7),\n\\beq\nN_0=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;f(p).\n\\eeq\nSince\n\\beq \np=\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\frac{{\\bf p}^2}{Dp^0}f(p)\\equiv p(m),\n\\eeq\n\\beq\n\\rho =\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;p^0 f(p)\\equiv \\rho (m)\n\\eeq\nand\n\\beq\nN_0=\\int \\frac{d^D{\\bf p}}{(2\\pi )^D}\\;f(p)\\equiv N_0(m)\n\\eeq\nare the standard expressions for the pressure, energy density and particle number density in $1+D$ dimensions, respectively \n\\cite{HW,Act,1+D}, we have the following relations:\n\\beq\np=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;p(m),\n\\eeq\n\\beq\n\\rho =\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;\\rho (m),\n\\eeq\n\\beq\nN_0=\\frac{T_{\\triangle V}}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;N_0(m).\n\\eeq\nIt is seen in these relations that the manifestly covariant framework\nprovides {\\it a linear mass spectrum, independent of the dimensionality of spacetime}. In order to obtain the expressions for the basic thermodynamic quantities, one has to integrate the standard (on-shell) results over this spectrum within the range of the mass distribution. \n\nUsing the formulas (2.22)-(2.24), one can establish the uncertainty relation (1.10) for a narrow mass width around the on-shell value: as $m\\rightarrow m_c,$ \n\\beq\n\\int _{m_1}^{m_2}dm\\;m\\;f(m)\\rightarrow \\triangle m\\;m_c\\;f(m_c),\n\\eeq\nwhere $f(m)$ stands for each of the $p(m),\\rho (m),N_0(m),$ and $\\triangle m$ is the (infinitesimal) width of the mass distribution around $m_c.$ The requirement that the results for $p,\\rho $ and $N_0$ coincide with those of the usual on-shell theories implies $p=p(m_c),\\rho =\\rho (m_c),N_0=N_0(m_c)$ in Eqs. (2.22)-(2.24), which leads, with Eq. (2.25), to the relation\\footnote{In c.g.s. units, this relation has a factor $\\hbar \/c^2$ on the right hand side.}\n\\beq\nT_{\\triangle V}\\triangle m=2\\pi ,\n\\eeq\nthe case of equality in the relation (1.10),\nin agreement with the results obtained earlier in ref. \\cite{glim}.\nOne can understand this relation, up to a numerical factor, in terms of\nthe uncertainty principle (rigorous in the $L^2(R^4)$ quantum theory)\n$\\triangle E\\cdot \\triangle t \\geq 1\/2.$ Since the time\ninterval for the particle to pass the volume $\\triangle V$ (this smallest\nmacroscopic volume is bounded from below by the size of the wave packets)\n$\\triangle t\\simeq E\/M \\;\\triangle \\tau ,$ and the dispersion of $E$ due\nto the mass distribution is $\\triangle E\\sim m\\triangle m\/E,$ one obtains\na lower bound for $T_{\\triangle V}\\triangle m$ of order unity.\n\n\\section{Mass-proper time uncertainty relation}\nWe now wish to prove the relation (1.10) for the general case, not\nonly for the case of a narrow mass width as done in the previous\nsection. First, we note that we previously considered a relativistic\nensemble without degeneracy; therefore no degeneracy factor appeared\nin the expressions for the basic thermodynamic quantities. Now suppose\nthat we have $\\nu$ internal degrees of freedom in our ensemble which correspond\nto degeneracy. In this case considerations made previously will remain valid and lead to the formulas (2.22)-(2.24) with the extra factor of $\\nu$ on their right hand side:\n\\beq\np=\\frac{T_{\\triangle V}\\nu}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;p(m),\n\\eeq\n\\beq\n\\rho =\\frac{T_{\\triangle V}\\nu}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;\\rho (m),\n\\eeq\n\\beq\nN_0=\\frac{T_{\\triangle V}\\nu}{2\\pi m_c}\\int _{m_1}^{m_2}dm\\;m\\;N_0(m).\n\\eeq\nOn the other hand, according to our previous arguments,\none can consider the $\\nu$ degrees of freedom as being distributed in the mass interval $m_1\\leq m\\leq m_2$ with a linear mass spectrum, \n\\beq \n\\tau (m)=Cm,\n\\eeq\nwhich leads to the formula\n\\beq\np=\\int _{m_1}^{m_2}dm\\;\\tau (m)\\;p(m),\n\\eeq\nand analogous relations for $\\rho $ and $N_0$ (similar to the\ntreatment of a strongly interacting system by means of a particle\nresonance spectrum \\cite{Shu}). In fact, the linear mass spectrum finds its confirmation in the experimental hadronic resonance spectrum: if one calculates, for example, the pressure in the hadronic resonance gas by summing up the individual contributions of a finite number of the different hadronic species with the corresponding degeneracies, $$p=\\sum _ig_i\\;p(m_i),\\;\\;\\;p(m_i)=\\frac{T^2m_i^2}{2\\pi ^2}K_2(m_i\/T),$$ and by using the formula (3.5), in which $m_1$ and $m_2$ are the masses of the lightest and the heaviest species, respectively, the results coincide \\cite{linear}.\n\nThe normalization constant $C$ is determined by the condition\n\\beq\n\\int _{m_1}^{m_2}\\tau (m)\\;dm=\\nu;\n\\eeq\ntherefore \n\\beq\nC=\\frac{2\\nu}{m_2^2-m_1^2}=\\frac{\\nu}{\\triangle m\\;(m_1+m_2)\/2},\n\\eeq\nwhere $\\triangle m=m_2-m_1$ is the width of the mass distribution in a general case. Since $m_c$ should be associated with one of the averages $\\langle m\\rangle $ or $\\langle m^2\\rangle $ which both are closer to $m_2$ than to $m_1$ for a linear spectrum [$(m_1+m_2)\/2\\stackrel{<}{\\sim } m_c$], \n\\beq\nC\\stackrel{>}{\\sim }\\frac{\\nu}{m_c\\;\\triangle m}.\n\\eeq\n\nDirect comparison of the formulas (3.1)-(3.3) with the relation (3.5) and analogous formulas for $\\rho $ and $N_0,$ with $\\tau (m)$ given by (3.4), \n\\beq\np=C\\int _{m_1}^{m_2}dm\\;m\\;p(m),\n\\eeq\n\\beq\n\\rho =C\\int _{m_1}^{m_2}dm\\;m\\;\\rho (m),\n\\eeq\n\\beq\nN_0=C\\int _{m_1}^{m_2}dm\\;m\\;N_0(m),\n\\eeq\nleads to the relation $$C=\\frac{T_{\\triangle V}\\nu}{2\\pi m_c},$$ which reduces,\nthrough (3.8), to\n\\beq\nT_{\\triangle V}\\triangle m\\stackrel{>}{\\sim }2\\pi ,\n\\eeq \nthe relation (1.10) for the general case of the finite range mass distribution.\n\nIn order that our considerations be valid, the effective degeneracy in\na relativistic ensemble should be large, $\\nu >>1,$ so that one could\nspeak of the distribution of $\\nu$ degrees of freedom in a finite mass\nrange. This is the case for realistic physical systems such as high temperature strongly interacting hadronic matter \\cite{Shu}. \n\n\\section{Concluding remarks}\nIn this paper we have proved the uncertainty relation (1.10) for the\ngeneral case of a finite range of mass distribution in a relativistic\nensemble. This relation allows one to admit the following picture of a\nstrongly interacting system: one can consider a strongly interacting\nsystem as a distribution of free particles which temporarily go off-shell\nwhile undergoing an interaction. Then $T_{\\triangle V}$ may be\nassociated with the time for the particle to remain very close to its mass shell. So, for weakly interacting systems, $\\triangle m\\simeq 0,$ $T_{\\triangle\nV}\\simeq \\infty ,$ i.e., the particle remains on-shell almost all the\ntime. In contrast, for a strongly interacting system, $\\mu _K\\simeq 0$\n\\cite{MS}; then, in view of (2.5), $\\triangle m\\simeq \\infty ,$ and\n$T_{\\triangle V}\\simeq 0,$ i.e., the particle is off-shell almost\nalways (because it undergoes interaction almost continuously).\n\nWe have remarked that the non-relativistic limit of the mass-``proper\ntime'' uncertainty relation provides a new derivation of the $\\triangle\nE \\triangle t \\stackrel{<}{\\sim } \\hbar $ relation of the\nnon-relativistic theory. This result implies, as we pointed out, the\nexistence of a residual ensemble over $t$, even in the\nnon-relativistic limit. Such an ensemble has been introduced recently\n\\cite{lax} in order to achieve an exact semigroup (exponential decay)\nlaw of evolution for the reduced motion of an unstable system. The\nusual derivation\\cite{jammer} of the non-relativistic energy-time uncertainty\nrelation studies the motion of the system under the action of the full\nHamiltonian relative to the eigenstates of unperturbed energy; this\nprocedure corresponds precisely to that of Weisskopf and\nWigner\\cite{wigner} for the description of the decay of unstable\nsystems. It is argued in refs. \\cite{lax} that the introduction of an\nensemble over $t$ is necessary for achieving the semigroup property as\nwell as for the consistency of the interpretation.\n\n\nThe uncertainty relation (1.10) may be useful in practical calculations concerning realistic strongly interacting systems in which the particles are necessarily off-shell. For example, it may allow one to estimate the relaxation times for the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.\n\n\\bigskip\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nDue to the strong field dynamics, critical phenomena arises in gravitational collapse in the threshold of black hole formation. In a similar way to phase transitions in thermodynamics, taking the mass of black hole as an order parameter, critical gravitational collapse can be classified as type I or type II. In type I collapse the final black hole mass has a minimum finite value, whereas in type II collapse the black hole mass can be arbitrary small.\n\t\nHistorically, M. Choptuik discovered a critical phenomena in gravitational collapse while studying numerically the gravitational collapse of a real massless scalar field, a result which was later named critical gravitational collapse of type II~\\cite{PhysRevLett.70.9}. He found that for a family of initial data parametrized by some arbitrary parameter $p$, the scalar field is completely dispersed to infinity for $pp^*$ the final black hole mass follows a power-law scaling relation of the form:\n\\begin{equation}\nM \\propto (p-p^*)^\\gamma \\; .\n\\end{equation}\nThe critical solution $p=p^*$ that separates both states exhibits universality, {\\em i.e.} it does not depend on the way in which the family is parameterized. Additionally, for different types of matter the critical solution can have either continuous self-similarity (CSS), or discrete self-similarity (DSS). In particular, for the case of a real massless scalar field the critical solution wa found to have DSS. This property is best appreciated in a logarithmic time defined as: \n\\begin{equation}\nT = -\\ln ( \\tau^*-\\tau ) \\; ,\n\\end{equation}\nwith $\\tau$ some measure of time which is usually taken as the proper time at the origin, and $\\tau^*$ the so-called accumulation time. In this logarithmic time $T$ the solution is periodic with period $\\Delta$. This property is known as ``scale echoing''. For the real massless scalar field the critical exponents have been found to be $\\gamma \\approx 0.374$ and $\\Delta\\approx3.445$, via both numerical simulations and semi-analytical studies \\cite{PhysRevLett.70.9,Rinne:2020asi,PhysRevD.49.890,PhysRevD.51.5558,Hamad__1996,PhysRevD.92.084037}.\n \nOn the other hand, critical collapse of type I was later discovered by Choptuik et. al. \\cite{PhysRevLett.77.424} while studying the critical collapse of a Yang-Mills field. In contrast to the critical collapse of type II, in this case the final black hole mass has a minimum finite value, and there is a different scaling law of the form:\n\\begin{equation}\n\\label{eq:tau_scaling}\n\\tau \\propto -\\gamma \\ln|p-p^*| \\; ,\n\\end{equation}\nwhere $\\tau$ now measures the time that a given solution remains near the critical solution. Additionally, the critical solution itself is either stationary or periodic in time.\n \nOne can expect type I critical collapse when in the field equations there exists either a mass or a length scale that is relevant to the dynamics. On the contrary, when the equations do not contain a length scale, or when such a length scale is not relevant, type II phenomena occur. Both types can coexists in different regions in the parameter space of initial data, as for example in the case of a real massive scalar field~\\cite{PhysRevD.56.R6057}, where the type of critical phenomena depends on the size of the Compton wavelength of the field when compared to the width of initial data (an excellent review about these type of critical phenomena can be found in~\\cite{Gundlach:2007gc}).\n \nIn this paper we study the critical collapse of a massive complex scalar field. A previous study was done by Hawley and Choptuik in \\cite{PhysRevD.62.104024}, showing that the critical solution for the complex scalar field corresponds to stationary solutions to the Einstein-Klein-Gordon known in the literature as boson stars~\\cite{PhysRev.172.1331,PhysRev.187.1767,Visinelli:2021uve,Liebling:2012fv}. These solutions are determined by assuming spherical symmetry, and by the requirement that the metric coefficients must be static, while the complex scalar field has a harmonic time dependence of the form:\n\\begin{equation}\\label{eq:BS_ansatz}\n\\Phi(t,r) = \\varphi(r)e^{i\\omega t} \\; ,\n\\end{equation}\nwith $\\omega$ a real valued frequency, and $\\varphi(r)$ a real valued function of radius only. Taking the mass parameter of the complex scalar field as $m$, the maximum possible mass of a boson star has been found to be $M_{max}\\approx0.633 \\: M_{Planck}^2\/m$, corresponding to a central value of the scalar field of $\\varphi_{max} = \\varphi(0) \\approx 0.271$, see for example~\\cite{PhysRevD.42.384,GLEISER1989733} (though this value can change depending on the normalization, see below for our normalization choice). This central value of the field separates the boson star configurations into two branches depending on their stability properties. If $\\varphi(0)<\\varphi_{max}$ the boson star is stable under small perturbations, whereas for $\\varphi(0)>\\varphi_{max}$ the configurations are unstable. The lowest energy solution for a boson star for a given value of $\\varphi(0)$, also known as the ground state, has no nodes in the scalar field. Excited states are classified depending on the number nodes of the field in the radial direction.\n\nThe critical solutions found by Hawley and Choptuik corresponded to unstable boson stars in the ground state, and were obtained by perturbing a stable boson star that interacted gravitationally with a small pulse of a massless scalar field that acts as the perturbation. In our study we take a different approach, and we begin with a simple gaussian pulse in the complex field with a variable width. We then evolve this initial data and vary the amplitude of the gaussian pulse until a critical solution is found. \n \nSince the mass of the complex scalar field introduces a scale, we expect our system to display both types of critical phenomena depending on the width of the initial gaussian pulse. In a similar way as in the case of a real massive scalar field~\\cite{PhysRevD.56.R6057}, we will explore both types of critical behaviour by changing the width of our initial pulse. Furthermore, if one performs a linear perturbative analysis for critical phenomena of both type I and II, the critical exponent $\\gamma$ can be shown to be the inverse of the so-called Lyapunov exponent $\\chi$ of the system $\\gamma=1\/\\chi$~\\cite{Gundlach:2007gc}.\\footnote{The Lyapunov exponent measures the stability of a system due to changes in its initial conditions. For close trajectories in phase space parametrized by $t$, and initial points separated by an infinitesimal distance $\\delta$, the Lyapunov exponent quantifies their rate of separation as $F(t,x_0+\\delta)-F(t,x_0)\\approx\\delta e^{\\chi t}$, with $\\chi$ the Lyapunov exponent of the system. For $\\chi>0$ the trajectories diverge, whereas for $\\chi<0$ they do not.}\n\nFor the case of critical collapse of type I, we will compare our critical solutions with the known solutions for stationary boson stars. Furthermore, we can also compare our critical exponents with the Lyapunov exponents for the unstable modes of boson stars. On the other hand, if the critical phenomena of type II we will limit our study to finding the critical exponent $\\gamma$ and the echoing period $\\Delta$.\n \nThis paper is organized as follows. In Section~\\ref{sec:scalarfield} we discuss the field equations for a complex massive scalar field and our initial data. In Section~\\ref{sec:code} we discuss our numerical code, gauge conditions and diagnostic tools. Section~\\ref{sec:results} show the results of our numerical simulations.\n\n\n\n\\section{Complex scalar field}\n\\label{sec:scalarfield}\n\n\n\n\\subsection{The Einstein Klein-Gordon equations}\n\nOur matter model consists of a massive complex scalar field $\\Phi$ coupled minimally to gravity, which can be described by the action (in units such that $G=c=1$):\n\\begin{equation}\nS = \\int d^4x \\sqrt{-g} \\left[ \\frac{R}{16\\pi}\n- \\frac{1}{2} \\left( \\nabla^\\mu\\Phi\\nabla_\\mu \\Phi^*\n+ m^2\\Phi\\Phi^* \\right) \\right] \\; ,\n\\end{equation}\nwhere $R$ is the Ricci scalar of the spacetime and $m$ is the mass parameter of the field (notice that this fixes our normalization choice). Varying the action with respect to the metric and the scalar field one obtains the Einstein field equations:\n\\begin{equation}\nR_{\\mu \\nu} - \\frac{1}{2} g_{\\mu \\nu} R = 8 \\pi T_{\\mu \\nu} \\; ,\n\\end{equation}\nand the Klein--Gordon equation:\n\\begin{equation}\\label{eq:KG}\n\\nabla^{\\mu} \\nabla_{\\mu} \\Phi-m^{2} \\Phi=0 \\; ,\n\\end{equation}\nwhere the stress-energy tensor $T_{\\mu \\nu}$ for the scalar field is given by:\n\\begin{equation}\nT_{\\mu\\nu} = \\frac{1}{2} \\left[ \\left( \\nabla_{\\mu} \\Phi \\nabla_{\\nu} \\Phi^{*}\n+ \\nabla_{\\nu} \\Phi \\nabla_{\\mu} \\Phi^{*} \\right)\n- g_{\\mu\\nu} \\left( \\nabla^\\alpha \\Phi \\nabla_\\alpha \\Phi^* + m^2 \\Phi \\Phi^* \\right) \\right] \\; .\n\\end{equation}\n\nIn order to study numerically the evolution of the system, we will use the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of general relativity \\cite{PhysRevD.52.5428,PhysRevD.59.024007}, which is known to be strongly hyperbolic \\cite{PhysRevD.66.064002}. Particularly, as we are only interested in the case of spherical symmetry, we will use the BSSN formulation adapted to curvilinear coordinates as described in~\\cite{PhysRevD.79.104029,Alcubierre:2011pkc}. In spherical symmetry, we will adopt the line element given by:\n\\begin{equation}\n\\label{eq:ds4}\nds^2 = -\\alpha^2 dt^2 + \\psi^4 \\left( A dr^2 + r^2 B d \\Omega^2 \\right) \\; ,\n\\end{equation}\nwhere $(\\alpha,\\psi,A,B)$ are functions of $(t,r)$ only, and $d\\Omega^2=d\\theta^2+\\sin^2\\theta\\ d\\varphi^2$ is the standard solid angle element.\n\nIn order to recast the Klein--Gordon equation as a first order system we define the following auxiliary variables\n\\begin{equation}\n\\Pi := \\frac{\\partial_t\\Phi}{\\alpha} \\; , \\qquad \\chi := \\partial_r \\Phi \\; .\n\\end{equation}\nWith these definitions, the Klein--Gordon equation \\eqref{eq:KG} can be rewritten as:\n\\begin{eqnarray}\n\\partial_t\\Phi &=& \\alpha\\Pi \\; , \\\\\n\\partial_t\\chi &=& \\alpha \\partial_r \\Pi + \\Pi \\partial_r \\alpha \\; , \\\\\n\\partial_t\\Pi &=& \\frac{\\alpha}{A\\psi^4} \\left[ \\partial_{r} \\chi\n+ \\chi \\left( \\frac{2}{r} - \\frac{\\partial_{r} A}{2 A} + \\frac{\\partial_{r} B}{B}\n+ 2 \\partial_{r} \\ln \\psi \\right) \\right]\n+ \\frac{\\chi \\partial_{r} \\alpha}{A \\psi^{4}} + \\alpha K \\Pi - \\alpha m^2 \\;L ,\n\\end{eqnarray}\nwith $K:=K^m_m$ the trace of the extrinsic curvature of the spatial hypersurfaces of constant time.\n\nFrom the orthogonal decomposition of the stress-energy tensor:\n\\begin{equation}\nT^{\\mu \\nu} = S^{\\mu \\nu} + J^{\\mu} n^{\\nu} + n^{\\mu} J^{\\nu} + \\rho n^{\\mu} n^{\\nu} \\;,\n\\end{equation}\nwe obtain the energy density $\\rho:=n^{\\mu} n^{\\nu} T_{\\mu \\nu}$, the momentum density $J_\\mu:= -P_{\\ \\mu}^{\\nu} n^{\\lambda} T_{\\nu \\lambda}$, and the stress tensor \\mbox{$S_{\\mu \\nu}:=P_{\\mu}^{\\sigma} P_{\\nu}^{\\lambda} T_{\\sigma \\lambda}$}, where $n^\\mu=(1\/\\alpha,0,0,0)$ is the unit normal vector to the spatial hypersurfaces and $P^\\mu_{\\ \\nu}:=\\delta^\\mu_{\\ \\nu}+n^\\mu n_\\nu$ is the projection operator. For the complex scalar field we find in particular:\n\\begin{eqnarray}\n\\rho &=& \\frac{1}{2} \\left( |\\Pi|^{2} + \\frac{|\\chi|^{2}}{A \\psi^{4}} + m^2|\\Phi|^2 \\right) \\; , \\\\\nJ_r &=& -\\frac{1}{2} \\left( \\rule{0mm}{5mm} \\chi \\Pi^{*} + \\Pi \\chi^{*} \\right) \\; , \\\\\nS_r^r &=& \\frac{1}{2} \\left( |\\Pi|^{2} + \\frac{|\\chi|^{2}}{A \\psi^{4}} - m^2|\\Phi|^2 \\right) \\; , \\\\\nS_{\\theta}^{\\theta} &=& \\frac{1}{2} \\left( |\\Pi|^{2} - \\frac{|\\chi|^{2}}{A \\psi^{4}}\n- m^{2}|\\Phi|^{2} \\right) \\; ,\n\\end{eqnarray}\n\n\n\n\\subsection{Initial data}\n\nIn \\cite{PhysRevD.62.104024}, Hawley and Choptuik showed that the critical solution for the cas of a massive complex scalar field is an unstable boson star. In their study the critical solution was obtained by perturbing a boson star in the stable branch with a real massless scalar field, and tuning the amplitude of the massless field up to the threshold of black hole formation. Here we will take a different approach, by considering as our initial condition a simple pulse of complex scalar field with the following gaussian profile:\n\\begin{eqnarray}\n\\Phi(t=0,r) &=& \\Phi_0 e^{-r^2\/\\sigma^2} \\label{eq:phi} \\; , \\\\\n\\Pi(t=0,r) &=& i \\kappa \\Phi_0 e^{-r^2\/\\sigma^2} \\label{eq:pi} \\; ,\n\\end{eqnarray}\nwhere $\\Phi_0,\\sigma,\\kappa$ are real parameters, and $\\Phi_0$ is the tuning amplitude of the initial pulse. In order to find the initial data for the geometry we assume a conformally flat spatial metric, so that $A=B=1$ in~\\eqref{eq:ds4}, and proceed to solve the constraint equations. Notice that even though equations \\eqref{eq:phi} and \\eqref{eq:pi} do not formally represent an instant of time symmetry, the momentum density $J_r$ is still zero, so the momentum constraint is trivially satisfied. On the other hand, at $t=0$ the hamiltonian constraint becomes a nonlinear second order differential equation for conformal factor $\\psi$ of the form:\n\\begin{equation}\n\\partial^2_r\\psi + \\frac{2}{r} \\: \\partial_r \\psi + 2 \\pi \\psi^5 \\rho = 0 \\; ,\n\\label{eq:ham_const}\n\\end{equation}\nwith the energy density given by:\n\\begin{equation}\n\\rho = \\frac{1}{2} \\left[ | \\Pi |^2 + \\frac{|\\partial_r \\Phi|^2}{\\psi^4} + m^2 |\\Phi|^2 \\right] \\; . \n\\end{equation}\nThe above non-linear equation is solved numerically by using an iterative method. Boundary conditions for equation \\eqref{eq:ham_const} are obtained from the asymptotically flatness condition, which implies:\n\\begin{equation}\n\\psi(r)|_{r\\rightarrow\\infty} = 1 \\; .\n\\end{equation} \nIn practice, however, we use a Robin boundary type condition at a finite radius corresponding to the edge of our numerical grid:\n\\begin{equation}\n\\partial_r\\psi = \\frac{1-\\psi}{r} \\; .\n\\end{equation}\nThis condition reflects the fact that as $r\\rightarrow\\infty$ we have $\\psi\\rightarrow1+{\\cal O}(r^{-1})$. On the other hand, regularity at the origin implies that $\\psi$ must be an even function of $r$, so that:\n\\begin{equation}\n\\partial_r \\psi|_{r=0} = 0 \\; .\n\\end{equation}\nThe initial conditions given by equations~\\eqref{eq:phi}-\\eqref{eq:pi} were not chosen randomly. At $t=0$ they are similar to the harmonic ansatz for boson stars, Eq.~\\eqref{eq:BS_ansatz}, with $\\kappa$ taking the role of the oscillation frequency $\\omega$ of the boson star, but instead of using the profile for a stable boson star we use a simple gaussian profile. Notice that $\\kappa$ is a free\nparameter, which we later choose to be equal to the mass $m$ of the scalar field for simplicity.\n\n\n\n\\section{Numerical code and diagnostics}\n\\label{sec:code}\n\n\\subsection{Code and gauge choices}\n\nWe integrate the Einstein--Klein--Gordon system with the OllinSphere code, a numerical relativity finite-difference code suited for spherical symmetry which evolves the BSSN formulation of the Einstein equations. This code has been previously used for example in~\\cite{Alcubierre:2019qnh,Degollado:2020lsa}, but it has now been updated to include the possibility of fixed mesh refinement around the origin, so that if the outer boundary is located at $r_{max}$ with grid resolution $\\Delta r$, the local boundary of the $N$ refinement level is situated at $r_{max}\/2^{N-1}$ with resolution $\\Delta r\/2^{N-1}$, and the grid structure remains fixed during the evolution.\n\nTo close the system we also need to specify the lapse function $\\alpha$ and the shift vector $\\beta$. In our simulations we choose for the lapse the standard $1+\\log$ slicing condition:\n\\begin{equation}\n\\partial_t\\alpha = -2 \\alpha K \\; ,\n\\end{equation}\nwith $K$ the trace of the extrinsic curvature. We choose for the initial value of the lapse a pre-collapsed profile of the form $\\alpha(t=0)=\\psi^{-2}$, with $\\psi$ the initial conformal factor.\nAlso, for simplicity we choose a vanishing shift vector for all our simulations. Indeed, this choice was already assumed in the line element~\\eqref{eq:ds4}.\n\n\n\n\\subsection{Diagnotics}\n\nAs is usual in the study of critical behavior, the final state of the evolution is classified depending on the strength of the initial data. We report our initial amplitude precision in finding the critical solution via the dimensionless quantity: \n\\begin{equation}\n\\delta\\Phi = \\frac{\\Phi_c - \\Phi_d}{\\Phi_d} \\; ,\n\\end{equation}\nwhere $\\Phi_c$ is the highest amplitude for which the initial data is dispersed and leaves behind Minkowski spacetime, and $\\Phi_d$ is the lowest amplitude for which a black hole is formed. The critical value for the amplitude $\\Phi^*$ is a metastable state which separates the two behaviors described previously. We have found that in order to obtain the critical exponents correctly we need an accuracy of at least $\\delta\\Phi\\sim10^{-6}$, o even higher. This value can be improved by using a finer grid, and results in less uncertainty in the value of the critical exponents, and also in a longer evolution near the critical solution for type I critical collapse.\n\nSince we are mostly interested in the subcritical case, for the supercritical simulations we will not follow the evolution until the black hole settles down to equilibrium, which in any case would require a non-zero shift vector. With our slicing choice, the lapse function at the origin will return to one if the initial scalar field pulse is dispersed to infinity. Otherwise, if a black hole is formed, the lapse will collapse to zero at the origin.\n\nIn order to detect when a black hole is formed we will search at every time step for an apparent horizon. This procedure is done by looking for a location where the expansion of outgoing null geodesics becomes zero (see for example~\\cite{Alcubierre:1138167}):\n\\begin{equation}\n\\frac{1}{\\psi^{2} \\sqrt{A}} \\left( \\frac{2}{r} + \\frac{\\partial_{r} B}{B}\n+ 4 \\frac{\\partial \\psi}{\\psi} \\right) - 2 K_{\\theta}^{\\theta} = 0 \\; .\n\\end{equation}\nHere $K_\\theta^\\theta$ is simply the angular component of the extrinsic curvature with mixed indices.\n\n\n\n\\subsection{Characterizing type I critical solutions}\n\nIn Eq.~\\eqref{eq:tau_scaling} we take $\\tau$ as the proper time measured by an observer located at the origin $r=0$ at a point in the evolution when a first apparent horizon is located. As explained before, the critical solutions of type I for the complex scalar field should correspond to an unstable boson star. We should emphasize, however, that our initial conditions for the complex scalar field given by Eqs.~\\eqref{eq:phi}-\\eqref{eq:pi} with the critical amplitude $\\Phi_0^*$ do not correspond to a boson star at $t=0$, unstable or otherwise. This implies that for our near critical simulations the excess of scalar field will be radiated to infinity and the remaining content should approach a boson star in the unstable branch.\n\nSince boson stars do not have a well defined boundary, one can describe their size by means of the so-called $R_{95}$ or $R_{99}$ radius, which correspond to the areal radius of a sphere containing $95\\%$ or $99\\%$ of the total mass $M_T$, respectively. Furthermore, since the system is not stationary the integrated mass will be a function of $M(t,r)$. To determine if a compact object has formed, we inspect the compactness function defined as:\n\\begin{equation}\\label{eq:compactness}\nC(t,r) = \\frac{M(t,r)}{R(t,r)} \\; ,\n\\end{equation}\nwhere $R(t,r)$ is the areal radius of a sphere at a given time $t$ and coordinate radius $r$. We look for the global maximum as a function of $r$ for every time step. We expect that if a boson star has formed, there will be a maximum mean value of $C(t,r)$, plus some small oscillations around it corresponding to perturbations of this star. This behavior will tell us if a compact object has formed or not, and will also provide us with an approximate lifetime of the critical solution obtained.\n\nIn Eq.~\\eqref{eq:compactness} we estimate the mass of the configuration by using the Kodama mass~\\cite{Kodama:1979vn, PhysRevD.91.084057,PhysRevD.96.064047,Racz:2005pm}, which is a quasi-local conserved energy in a spherically symmetric spacetime. The Kodama vector is defined by:\n\\begin{equation}\nK^A = \\epsilon^{AB} \\partial_{B} R \\; ,\n\\end{equation}\nwhere $R$ is the areal radius of a sphere at constant $t$ and $r$, $\\epsilon^{A B}$ is the totally antisymetric tensor in the two-dimensional manifold with coordinates $(t,r)$, and the indices $(A,B)$ run over $(0,1)$. The vector $K^A$ can be naturally extended to the four dimensional manifold \nby setting to zero the remaining components. Next, we define the four vector $S^\\mu$ as follows:\n\\begin{equation}\n{\\cal{S}}^\\mu = T^{\\mu \\nu} K_\\nu \\; ,\n\\end{equation}\nwhere $T^{\\mu \\nu}$ is the stress-energy tensor. It is possible to show that $S^\\mu$ is a conserved current, so it satisfies the conservation law:\n\\begin{equation}\n\\partial_\\mu \\left( \\sqrt{-g} {\\cal S}^\\mu \\right) = 0 \\; ,\n\\end{equation}\nIn a sphere of radius $r_0$ at constant $t$, we can then define a conserved mass (the so-called Kodama mass) as:\n\\begin{equation\nM\\left(t, r_{0}\\right) := \\int_{\\text {sphere }} S^{t} \\alpha \\sqrt{\\gamma} \\:d x^{3} \\;,\n\\end{equation}\nwhere $\\gamma$ is the determinant of the 3-metric, and where we used the fact that $-g = \\alpha \\gamma$. Using our expression for the spatial metric this reduces to:\n\\begin{equation}\\label{eq:M_Kodama}\nM(t r_{0}) := 4 \\pi \\int_0^{r_0} \\alpha {\\cal S}^{t} r^2 \\psi^6 A^{1\/2} B \\: dr \\; ,\n\\end{equation}\nOn the other hand, from the above definitions the quantity $S^t$ can now be expressed as:\n\\begin{equation}\n{\\cal S}^t = \\frac{r}{\\alpha} \\sqrt{\\frac{B}{A}} \\left[ \\rho \\left( 2 r \\chi\n+ \\frac{\\partial_r B}{2 B} + \\frac{1}{r} \\right)\n- J_r K_\\theta^\\theta\\right] \\; .\n\\end{equation}\nIn a similar way, one can also define the integrated energy flux through the sphere of radius $r_0$ as:\n\\begin{equation}\n\\label{eq:P_Kodama}\nP(t_1,t_2;r_0) := 4 \\pi \\int_{t_1}^{t_2} \\left. \\left( \\alpha {\\cal S}^r \\sqrt{\\gamma} \\right) \\right|_{r_0} dt \\; ,\n\\end{equation}\nwhere ${\\cal S}^r$ is now given by: \n\\begin{equation}\n{\\cal S}^r = r \\sqrt{\\frac{B}{A}} \\left[ \\frac{1}{\\psi^4A} \\left( 2 r \\chi\n+ \\frac{\\partial_r B}{2B} + \\frac{1}{r} \\right)\n- S_r^r K_\\theta^\\theta \\right] \\; .\n\\end{equation}\nWith the definitions \\eqref{eq:M_Kodama} and \\eqref{eq:P_Kodama}, the conservation law can be rewritten as:\n\\begin{equation}\n\\partial_t \\left[ M(t,r_0) + P(t_0,t;r_0) \\right] = 0 \\; .\n\\end{equation}\n\nWe still need to estimate the integration radius $r_0$. In order to do this we use the fact that stationary boson stars are well characterized in the literature, and our code is able to find such solutions (see for example~\\cite{Alcubierre:2018ahf}). For a near critical simulation we then first obtain the mean value of the scalar field amplitude at the origin, $\\left< \\Phi(t,r=0) \\right>$. Having found this mean amplitude, we construct the corresponding boson star stationary solution with that same amplitude, and choose $r_0$ as the $R_{99}$ radius of that stationary solution.\n\n\n\n\\subsection{Characterizing type II critical collapse}\n\nAs the mass of the scalar field introduces a length (or mass) scale, following \\cite{PhysRevD.56.R6057} we expect that if $\\sigma m \\ll 1$ we should observe critical collapse of type II, whereas if $\\sigma m \\gg 1$ we should find collapse of type I. Finding the value of the critical exponent $\\gamma$ using the black hole mass scaling is somewhat difficult since we would need to follow the black hole until it reaches an equilibrum configuration, something that is not trivial to do numerically. Instead, we will consider subcritical evolutions since in a critical collapse of type II the maximum value of the 4D Ricci scalar will then follow the scaling law:\n\\begin{equation}\nR_{max} \\approx |\\Phi_0^*-\\Phi_0|^{-2\\gamma} \\; ,\n\\end{equation}\nwhere the $-2$ factor in the exponent is there because the Ricci scalar has units of lenght$^{-2}$. Additionally to this behavior, the discrete self-similarity of the phenomena adds a fine structure to the scaling law \\cite{PhysRevD.55.R440}, so the Ricci scalar in fact will behave as:\n\\begin{equation}\n\\label{eq:4DR_1}\n\\ln R_{max} = c - 2 \\gamma \\ln |\\Phi_0^* - \\Phi_0| + f( \\ln |\\Phi_0^*-\\Phi_0| ) \\; ,\n\\end{equation}\nwith $c$ some constant, and where $f$ is a periodic function with angular frequency:\n\\begin{equation}\n\\omega = \\Delta \/2 \\gamma \\; ,\n\\end{equation}\nwith $\\Delta$ is the so-called echoing period. To leading order, $f$ can be approximated by:\n\\begin{equation}\nf(x) = a_0 \\sin \\left( \\omega x + \\varphi \\right) \\; ,\n\\end{equation}\nwith $\\varphi$ some arbitrary phase. The 4D Ricci scalar then behaves as:\n\\begin{equation}\n\\label{eq:4DR_2}\n\\ln R_{max} = c - 2\\gamma \\ln |\\Phi_0^*-\\Phi_0|\n+ a_0 \\sin \\left( \\omega \\ln |\\Phi_0^*-\\Phi_0| + \\varphi \\right) \\; ,\n\\end{equation}\nwhere the constants $c,a_0,\\varphi$ depend on the form of the initial data family.\n\n\n\n\\section{Numerical results}\n\\label{sec:results}\n\nAll our simulations were done with fourth order centered differences in space, and fourth order Runge--Kutta for the evolution in time. For simplicity, we fix the scalar field mass to $m=1$. Also, for the family of initial data~\\eqref{eq:phi}-\\eqref{eq:pi} we set $\\kappa=m=1$ for all cases. The values chosen for the width parameter $\\sigma$ will be reported below. The parameter $\\Phi_0$ is then adjusted until we find the black hole formation threshold with the desired accuracy.\n\nTo reduce the source of errors in our simulations we use constraint preserving boundary conditions. These have already been described and used for example in~\\cite{Alcubierre:2014joa,PhysRevD.86.104044}, and they help to reduce the errors coming in from the boundaries by a factor of about $10^3$ when compared with the standard Sommerfeld (radiative) boundary conditions. The error introduced by the finite difference method can also be diminished by using Kreiss--Oliger numerical dissipation. In all our evolutions we use sixth-order dissipation in order to be compatible with the fourth-order discretization. The artificial dissipation dampens high frequency modes that would otherwise spoil the numerical stability of the near-critical solutions. Resolution also affects the critical behavior, for this reason we will report relevant quantities for our highest resolution simulations.\n\n\n\n\\subsection{Type II critical collapse}\n\nAs already stated before, we expect type II critical collapse for $\\sigma m \\ll 1$. To check this, we choose $\\sigma\\leq0.5$ and proceed to find the critical amplitude $\\Phi_0^*$ using a bisection method. Since studying critical phenomena requires high numerical precision, instead of using many levels of refinement, which introduce reflections at the refinement boundaries, we will use just one grid level with a radial transformation of coordinates from the original coordinate radius $r$ to a new coordinate $\\tilde{r}$ related to $r$ through:\n\\begin{equation}\n\\label{eq:rnew}\n\\frac{dr}{d\\tilde{r}} = \\frac{1}{1+e^{\\beta r^2+\\delta}} \\; .\n\\end{equation}\nWith this transformation a uniform grid in $\\tilde{r}$ becomes a non-uniform grid in $r$. This coordinate transformation was first used in~\\cite{PhysRevD.105.064071} for studying the critical behavior of scalar-tensor theories of gravity in the Jordan frame. In Eq. \\eqref{eq:rnew}, $\\delta$ adjusts the resolution near the origin $\\tilde{r}=0$, while $\\beta$ measures how fast $\\tilde{r}$ approaches $r$ far away. Notice that as the transformed radial coordinate approaches infinity $\\tilde{r}\\rightarrow\\infty$, we have $dr\/d\\tilde{r}\\rightarrow 1$. For our simulations we use $\\delta=5$ and $\\beta=-1$, with a grid spacing $\\Delta \\tilde{r}=0.005$, and $N_r=2500$ points in radial direction. We also use an adaptive time step in order to satisfy the Courant--Friedrichs--Levi (CFL) stability condition. With these settings we were able to find the critical amplitude with a precision of $\\delta\\Phi\\approx10^{-12}$.\n\nFigure~\\ref{fig:lnRmax} shows the maximum value of the 4D Ricci scalar at the origin obtained from subcritical evolutions for $\\sigma=0.5$. From the Figure we can clearly see the expected behaviour for type II critical collapse. All cases $\\sigma\\leq0.5$, fitting the function \\eqref{eq:4DR_2} results in very similar critical exponent $\\gamma \\approx 0.38 \\pm 0.01$ and echoing period $\\Delta \\approx 3.4\\pm 0.1$. These values are very similar to those found in the literature for the case of a real massless scalar field.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/lnRmax.pdf}\n\\caption{Scaling of the maximum value of the 4D Ricci scalar for subcritical simulations of a massive complex scalar field, using the initial data family \\eqref{eq:phi}-\\eqref{eq:pi}, with gaussian width $\\sigma=0.5$.}\n\\label{fig:lnRmax}\n\\end{figure}\n\n\n\n\\subsection{Type I critical collapse}\n\nAs already mentioned, for the case when $\\sigma m\\gg 1$ we expect to find type I critical collapse. To investigate this, we start from $\\sigma=2.5$ and consider higher values of the gaussian width. Once the value of $\\sigma$ has been chosen, for each case we proceed to find the critical amplitude $\\Phi_0^*$ with the bisection method. For these simulations we have used fixed mesh refinement with $N=4$ levels. We have observed that the lifetime of the nearest critical solution obtained initially increases as $\\sigma$ is increased, reaching its maximum at $\\sigma \\approx 4.0$, and then decreases again for higher values of $\\sigma$.\n\nTo set the position of the outer boundary, we first estimate the time of black hole formation $t_{BH}$ using low resolution runs. We then multiply $t_{BH}$ the asymptotic gauge speed $v_g=\\sqrt{2}$ of the $1+\\log$ slicing condition (as it is larger than the coordinate speed of light $v_l=1$). A perturbation that starts at the origin and bounces at the boundary will take at least twice this time to return to the origin (in fact longer since the lapse is smaller than one near the origin), so we set the outer boundary at $r_{max}=t_{BH}\/\\sqrt{2}$. The position of the outer boundary $r_{max}$, and the resolution of the base grid $\\Delta r$, are displayed in Table~\\ref{tab:resolution_bdry}. In all these simulations we use a fixed time step compatible with the CFL condition.\n\n\\begin{table}\n\\begin{tabular}{@{}|c|c|c|@{}}\n\t\\hline\n\t\\multicolumn{1}{|c|}{$\\sigma$} & \\multicolumn{1}{c|}{$\\Delta r$} & \\multicolumn{1}{c|}{$r_{max}$ } \\\\ \\hline\n\t2.5 & & 150 \\\\\n\t2.75 & 0.1 & 200 \\\\\n\t3 & & 250 \\\\ \\hline\n\t3.5 & & 400 \\\\\n\t4 & 0.15 & 550 \\\\\n\t5 & & 400 \\\\ \\hline\n\t6 & & 350 \\\\\n\t7 & & 325 \\\\\n\t8 & 0.1 & 325 \\\\\n\t9 & & 290 \\\\\n\t10 & & 225 \\\\ \\hline\n\\end{tabular}\n\\caption{Resolution and position of the outer boundary for each value of the width parameter $\\sigma$.}\n\\label{tab:resolution_bdry}\n\\end{table}\n\nFigure~\\ref{fig:compactness-norm} shows the maximum value of the compactness function (top panel), and the norm of the complex scalar field at the origin (bottom panel), for a near critical solution with initial width $\\sigma=2.5$. Since the initial profile consists of a gaussian pulse, the scalar field has not agglomerated to form a compact object. After $t \\approx 25$ a portion of the scalar field has been radiated away and the remainder starts to oscillate around a mean value, indicating that a compact object has been formed. As this state is unstable, the object eventually disperses after $t \\approx 225$. It is important to mention, however, that for some values of the initial amplitude $\\Phi_0$ the scalar field does not disperse completely, and the remaining bulk oscillates around the origin. This behavior is similar to that observed by Lai and Choptuik in~\\cite{Lai:2007tj}. In their study, the remaining bulk can be described as excitations of the fundamental mode of stable boson stars. However we will leave the study of this phenomenon for a future work.\n\nIn the time interval $25 < t < 225$, we obtain the mean value of the norm of the complex field at the origin, and then look for the $R_{99}$ of the corresponding boson star. Figure~\\ref{fig:KodamaMass} shows the Kodama mass for the same subcritical evolution measured at that radius. Again, we observe an oscillation around a mean value, and the dispersion of the object after some time. We can also obtain the oscillation frequency of the scalar field by applying a Fast Fourier Transform (FFT) to the central value of its real and imaginary parts. Figure~\\ref{fig:freq_sigma_2p5} plots the frequency obtained after applying the FFT. We can clearly see a very narrow peak centered at $\\omega=0.7933$.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/compactness_s=2.5.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/complex_phi_norm_s=2.5.pdf}\n\\caption{Top panel: Maximum value of the compactness function for a near critical evolution with $\\sigma=2.5$. Bottom panel: Norm of the scalar field at the origin. Initially we see no indication that a compact object has formed. However, from $t \\approx 25$ up to $t \\approx 225$ we can appreciate a clear oscillation around a mean value. Since this is a subcritical case, we see dispersion of the object for $t > 225$.}\n\\label{fig:compactness-norm}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/mass_s=2.5.pdf}\n\\caption{Kodama mass for the same simulation of Figure~\\ref{fig:compactness-norm}. After obtaining the mean value of the norm of the scalar field at the origin we find the $R_{99}$ of the corresponding boson star. We then evaluate the Kodama mass at that radius.}\n\\label{fig:KodamaMass}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/w-sigma_2p5.pdf}\n\\caption{Fourier transform of the central value of the scalar field for $\\sigma=2.5$. Both the real and imaginary parts have a very narrow peak centered at a frequency $\\omega=0.7933$.}\n\\label{fig:freq_sigma_2p5}\n\\end{figure}\n\nWe summarize the results for all our simulations in Table~\\ref{tab:all_data}, where we report the mean value of the norm of the scalar field at the origin, its oscillation frequenc\n, and the Kodama mass of the critical solution. The uncertainty in the norm of the complex field and its mass are calculated from the standard deviation, and the uncertainty in the frequency is reported as half of the peak width in the FFT. The uncertainty in the critical exponent is obtained from the method of least squares applied to equation~\\eqref{eq:tau_scaling}.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|}\n \t\t\\hline\n \t\t$\\sigma$ & $|\\bar{\\Phi}(r=0)|$ & $\\bar{M}$ & $\\omega$ & $\\gamma$ \\\\ \\hline\n \t\t2.5 & 0.127$\\pm$0.008 & 0.590 $\\pm$ 0.003 & 0.793 $\\pm$ 0.008 & 5.075 $\\pm$ 0.024 \\\\\n \t\t2.75 & 0.115$\\pm$0.007 & 0.606 $\\pm$ 0.002 & 0.804 $\\pm$ 0.006 & 6.884 $\\pm$ 0.022 \\\\\n \t\t3 & 0.106$\\pm$0.006 & 0.615 $\\pm$ 0.003 & 0.813 $\\pm$ 0.004 & 9.168 $\\pm$ 0.023 \\\\\n \t\t3.5 & 0.092$\\pm$0.004 & 0.629 $\\pm$ 0.002 & 0.833 $\\pm$ 0.008 & 15.592 $\\pm$ 0.042 \\\\\n \t\t4 & 0.086$\\pm$0.004 & 0.631 $\\pm$ 0.002 & 0.842 $\\pm$ 0.006 & 20.887 $\\pm$ 0.061 \\\\\n \t\t5 & 0.092$\\pm$0.003 & 0.628 $\\pm$ 0.002 & 0.832 $\\pm$ 0.008 & 15.494 $\\pm$ 0.023 \\\\\n \t\t6 & 0.098$\\pm$0.004 & 0.623 $\\pm$ 0.003 & 0.826 $\\pm$ 0.009 & 11.567 $\\pm$ 0.035 \\\\\n \t\t7 & 0.104$\\pm$0.004 & 0.621 $\\pm$ 0.003 & 0.820 $\\pm$ 0.011 & 9.572 $\\pm$ 0.022 \\\\\n \t\t8 & 0.108$\\pm$0.005 & 0.613 $\\pm$ 0.004 & 0.813 $\\pm$ 0.012 & 8.423 $\\pm$ 0.023 \\\\\n \t\t9 & 0.112$\\pm$0.006 & 0.609 $\\pm$ 0.004 & 0.810 $\\pm$ 0.014 & 7.608 $\\pm$ 0.016 \\\\\n \t\t10 & 0.115$\\pm$0.007 & 0.605 $\\pm$ 0.005 & 0.806 $\\pm$ 0.006 & 7.051 $\\pm$ 0.024 \\\\ \n \t\t\\hline\n\\end{tabular}\n\\caption{Summary of our numerical results for the critical solutions. Since $\\Phi(r=0)$ and $M$ have an oscillatory behavior, we report the mean value with an uncertainty given by the standard deviation. The frequency is obtained using a FFT applied to the real and imaginary parts of the field at the origin. The critical exponent $\\gamma$ is calculated using a least squares fit to eq.~\\eqref{eq:tau_scaling}. We notice that as $\\sigma$ increases, the mass of the critical solution first approaches the highest possible value for the mass for a boson star $M \\sim 0.633$, reaching the maximum value for $\\sigma=4.0$, while for higher values of $\\sigma$ the mass decreases again.}\n\\label{tab:all_data}\n\\end{table}\n\n\\vspace{5mm}\n\nWe now turn to the question of whether our critical solutions do indeed correspond to unstable boson stars as found Hawley and Choptuik in~\\cite{PhysRevD.62.104024}, even if our initial data is very different. As a first comparison, Figure \\ref{fig:BS-s4-10} shows the norm of the complex scalar field for our critical solution as a function of areal radius for the cases with $\\sigma=4$ (top pannel) and $\\sigma=10$ (bottom panel), superimposed with the norm of the complex scalar field for an unstable boson star with the same amplitude. We notice that the critical solutions obtained have no nodes in the field, so the corresponding boson star is in its ground state. We can clearly see that the profiles of our critical solutions follow very closely the expected profile\nfor the boson stars.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/BS-s4.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/BS-s10.pdf}\n\\caption{Comparison of the norm of the complex scalar field of our critical solutions with that of unstable boson stars with the same amplitude, for the cases $\\sigma=4$ (top pannel) and $\\sigma=10$ (bottom panel). The dots represents the critical solutions, and the solid lines the corresponding boson stars.}\n\\label{fig:BS-s4-10}\n\\end{figure}\n\nNext, in Figure~\\ref{fig:mass_freq} we show a plot of the mass vs. the frequency of oscillation for our critical solutions, corresponding to the data in table~\\ref{tab:all_data}, compared to the same plot for boson star solutions (solid lines). We separate the data into two plots to make more evident the fact that the mass of the critical solution first increases with $\\sigma$ up to $\\sigma=4$, and then decreases again with higher values of $\\sigma$. Figure~\\ref{fig:mass_phi} shows a similar plot but now of the mass vs. the central value of the norm of the scalar field. As can be seen in the plots, our critical solutions fall directly in the line for stationary boson stars. Moreover, they are all to the left of the maximum mass in Figure~\\ref{fig:mass_freq} and to the right of the maximum in Figure~~\\ref{fig:mass_phi}, which correspond to the unstable branch for boson stars. The critical solution for $\\sigma=4$ is almost at the maximum mass. Figure~\\ref{fig:scaling_tau} shows the scaling of $\\tau$, the lifetime of the near critical solutions for the different values of $\\sigma$. We can see that the scaling shows good agreement with Eq.~\\eqref{eq:tau_scaling}, but with different critical exponents for the different values of $\\sigma$.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/M-w_1.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/M-w_2.pdf}\n\\caption{Mass and oscillation frequency for our critical solutions compared with the curve for stationary boson star solutions. Circles corresponds to the specific value of $\\sigma$ and the solid line to the known values for stationary boson stars. Top panel: Critical solutions for $\\sigma \\leq 4$. Bottom panel: Critical solutions for $\\sigma \\geq 4$. We observe that the maximum value for the mass is reached for $\\sigma$ such that $3.5 < \\sigma < 5$.}\n\\label{fig:mass_freq}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/phi-M_1.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/phi-M_2.pdf}\n\\caption{Mass and scalar field norm at the origin for our critical solutions compared with the curve for stationary boson star solutions. Circles corresponds to the specific value of $\\sigma$ and the solid line to the known values for stationary boson stars. Top panel: Critical solutions for $\\sigma \\leq 4$. Bottom panel: Critical solutions for $\\sigma \\geq 4$.}\n\\label{fig:mass_phi}\n\\end{figure}\n\nSince the ADM mass of the initial pulse increases monotonically relative to $\\sigma$, we could have expected that the mass of the critical solution approaches asymptotically the maximum value for boson stars $M_{ADM} \\sim 0.633$, which separates the unstable from the stable regions, as $\\sigma$ is increased. But as can be seen from the plots and the data ofTable~\\ref{tab:all_data}, instead we find that for $\\sigma \\lesssim 4$ the mass of the critical solution increases, while for higher values of $\\sigma$ the mass decreases and moves away from the maximum mass value. Our data indicates that the maximum possible mass for a boson star will probably be attained for $\\sigma $ between $3.5< \\sigma <5$. This behavior is also reflected in the values of critical exponent $\\gamma$, which also reaches its maximum value between $3.5 < \\sigma < 5$.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/scaling_tau.pdf}\n\\caption{Scaling of the lifetime of near critical solutions for different values of the gaussian width $\\sigma$. With the values of $\\sigma$ tested, the maximum value of the critical exponent $\\gamma$ is reached at $\\sigma=4$. }\n\\label{fig:scaling_tau}\n\\end{figure}\n\nFinally, for boson stars the imaginary part of Lyapunov exponent $\\lambda$ can be related to the critical exponent $\\gamma$, by \\mbox{$\\text{Im}(\\chi) = 1\/\\gamma$} (details about the procedure to obtain the Lyapunov exponents can be consulted in~\\cite{PhysRevD.62.104024}). Figure~\\ref{fig:lyapunov} compares the square of the Lyapunov exponent for boson stars obtained through a linear perturbation analysis (data provided by A.~Bernal~\\cite{BernalA_2021}), with the critical exponents $1\/\\gamma^2$ measured in our simulations. We can see an excellent agreement between both data sets.\n\n\\begin{figure}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/Lyapunov_1.pdf}\n\\includegraphics[width=0.75\\textwidth]{.\/plots\/Lyapunov_2.pdf}\n\\caption {Comparison of the square of Lyapunov exponent for unstable modes of boson stars. Circles correspond to the $1\/\\gamma^2$ for the specific values of $\\sigma$, while the solid lines correspond to the Lyapunov exponents obtaned from a linear perturbation analysis of boson star solutions. Top panel: for $\\sigma \\leq 4$. Bottom panel: $\\sigma \\geq 4$.}\n\\label{fig:lyapunov}\n\\end{figure} \n\n\n\\section{Conclusions}\n\nWe performed numerical simulations of a massive complex scalar field using a numerical code adapted to spherical symmetry in order to study critical gravitational collapse. Our initial conditions for the complex field are somewhat similar to the harmonic boson star ansatz, but crucially they do not correspond to a stationary boson star solution.\n\nWe find that, depending on the width of initial data, the critical collapse behaves in two very different ways. For $\\sigma \\leq 0.5$ we can measure the 4D Ricci scaling, which is indicative of type II critical collapse. We obtain values for the critical exponent $\\gamma = 0.38 \\pm 0.01$ and echoing period $\\Delta = 3.4 \\pm 0.1$, which are very similar to those found in the literaure for the case of a real massless scalar field. On the other hand, for $\\sigma \\geq 2.5$ we obtain the scaling of the lifetime of near critical solutions, which is characteristic of type I critical collapse. For type I collapse we observe that the critical exponent depends on the initial gaussian width $\\sigma$: as this width increases the critical exponent reaches its highest value for $\\sigma \\approx 4$, while for higher values of $\\sigma$ the value of the critical exponent decreases again.\n\nIn a similar way to Hawley and Choptuik~\\cite{PhysRevD.62.104024}, we also find that the critical solutions obtained correspond to boson stars in the ground state in the unstable branch. We validate our results by contrasting the curves of $|\\Phi(0)|$ vs. $M_{ADM}$, and $\\omega$ vs $M_{ADM}$. Up to our uncertainties we find that our critical solutions do fall on the curves for unstable stationary boson stars. Also, for our simulations the maximum mass of the critical solution is obtained for $\\sigma \\simeq 4$, which leads us to conjecture that the maximum mass of a boson star at the boundary between the stable and unstable branches, $M \\sim 0.633$, will be attained for $\\sigma$ somewhere in the range $3.5<\\sigma<5.0$. Furthermore, we also confirm that the inverse of the critical exponent $\\gamma$ for our critical solutions does indeed correspond to the imaginary part of Lyapunov exponents for unstable boson stars obtained through a linear perturbation analysis.\n\nOne final comment about the transition from type I critical collapse to type II. Since we have obtained the two different behaviors by varying the value of $\\sigma$, we can in principle study the transition between both types of collapse by concentrating in the region $0.5 < \\sigma < 2.5$. On the other hand, as can be seen from our plots, for type I collapse as $\\sigma$ decreases the critical solution moves further into the unstable branch for boson stars. This raises the questions as to how far down this branch we can go before transitioning to type II critical collapse. We will leave this question for a future study.\n\n\n\n\\acknowledgments\n\nThis work was partially supported by CONACyT Network Projects No. 376127 and No. 304001. EJ was also supported by a CONACyT National Graduate Grant.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nDirected relations are essential to explaining pairwise dependencies among multiple interacting units. \nIn gene network analysis, regulatory gene-to-gene relations are a focus of biological investigation \\citep{sachs2005causal}, while in a human brain network, scientists investigate causal influences among regions of interest to understand how the brain functions through the regional effects of neurons \\citep{liu2017effective}. \nIn such a situation, inferring directed effects without other information is generally impossible because of the lack of identifiability in a Gaussian directed acyclic graph (DAG) model, and hence external interventions are introduced to treat a non-identifiable situation \\citep{heinze2018causal}. \nFor instance, the genetic variants such as single-nucleotide polymorphisms (SNPs) can be, and indeed are increasingly, treated as external interventions to infer inter-trait causal relations in a quantitative trait network \\citep{brown2020phenome} and gene interactions in a gene regulatory network \\citep{teumer2018common,molstad2021covariance}. \nIn neuroimaging analysis, scientists use randomized experimental stimuli as interventions \nto identify causal relations in a functional brain network \\citep{grosse2016identification,bergmann2021inferring}.\nHowever, the interventions in these studies often have unknown targets and off-target effects \\citep{jackson2003expression,eaton2007exact}. \nConsequently, inferring directed relations while identifying useful interventions for inference is critical.\nThis paper focuses on the simultaneous inference of directed relations subject to unspecified interventions, that is, without known targets.\n\nIn a DAG model, the research has been centered on the reconstruction of directed\nrelations in observational and interventional studies \\citep{van2013ell,oates2016estimating,heinze2018causal,zheng2018dags,yuan2019constrained}. \nFor inference, Bayesian methods \\citep{friedman2003being,luo2011bayesian,viinikka2020towards} have been popular. \nYet, statistical inference remains under-studied, especially for interventional models in high dimensions \\citep{peters2016causal,rothenhausler2019causal}.\nRecently, for observational data, \\citet{jankova2018inference} propose a debiased test of the strength of \na single directed relation and \\citet{li2019likelihood} derives a constrained likelihood ratio test for multiple directed relations.\n\nDespite progress, challenges remain. First, inferring directed relations requires identifying a certain DAG topological order \\citep{van2013ell}, while the identifiability in a Gaussian DAG with unspecified interventions remains under-explored. Second, the inferential results should agree with the acyclicity requirement for a DAG. As a result, degenerate and intractable situations can occur, making inference greatly different from the classical ones. Third, likelihood-based methods for learning the DAG topological order often use permutation search \\citep{van2013ell} or continuous optimization subject to the acyclicity constraint \\citep{zheng2018dags,yuan2019constrained}, \nwhere a theoretical guarantee of the actual estimate (instead of the global optimum) has not been established for these approaches.\n{Recently, an important line of work \\citep{ghoshal2018learning,rajendran2021structure,rolland2022score} has focused on order-based algorithms with computational and statistical guarantees. However, in Gaussian DAGs, existing methods often rely on some error scale assumptions \\citep{bulhmann2013inde}, which is sensitive to variable scaling like the common practice of standardizing variables. This drawback could limit their applications, especially in causal inference, as causal relations are typically invariant to scaling.}\n\n{\n To address the above issues, we develop structure learning and inference methods for a Gaussian DAG with unspecified additive interventions. Unlike the existing approach where structure discovery and subsequent inference are treated separately, our proposal integrates DAG structure learning and testing directed relations (also known as the natural direct effect in causal inference), accounting for the uncertainty of structure learning for inference. With suitable interventions called instrumental variables (IVs), the proposed approach removes the restrictive error scale assumptions and delivers creditable outcomes with theoretical guarantees in low order polynomial time. This result indicates IVs, a well-known tool in causal inference, can play important roles in structure learning even if some interventions do not meet the IV criterion.\n More specifically, our contributions are summarized as the following aspects.}\n\n\\begin{itemize}\n \\item For modeling, we establish the identifiability conditions for a Gaussian DAG with unspecified interventions. In particular, the conditions allow interventions on more than one target, which is suitable for multivariate causal analysis \\citep{murray2006avoiding}. Moreover, we introduce the concepts of nondegeneracy and regularity for hypothesis testing under a DAG model to characterize the behavior of a test.\n\n \\item For methodology, we develop likelihood ratio tests for directed edges and pathways in a super-graph of the true DAG, where the super-graph is formed by ancestral relations and candidate interventional relations, offering the topological order for inference. \nFurthermore, we reconstruct the super-graph by the peeling algorithm, which automatically meets the acyclicity requirement. {By integrating structure learning with inference, we account for the uncertainty of the super-graph estimation for the proposed tests via a novel data perturbation (DP) scheme, which effectively controls the type-I error while enjoying high statistical power.} \n\n \\item For theory, we prove that the proposed peeling algorithm based on nodewise regressions yields consistent results in {$O(p \\times \\log\\kappa_{\\max}^\\circ \\times (q^3 + nq^2) )$ operations almost surely, where $p,q$ are the numbers of primary and intervention variables, $n$ is the sample size, and $\\kappa_{\\max}^\\circ$ is the sparsity.} On this basis, we justify the proposed DP inference method by establishing the convergence of the DP likelihood ratio to the null distribution and desired power properties.\n \n \\item The numerical studies and real data analysis demonstrate the utility and effectiveness of the proposed methods. The implementation of the proposed tests and structure learning method is available at \\url{https:\/\/github.com\/chunlinli\/intdag}.\n\n\\end{itemize}\n\n{The rest of the article is structured as follows.\nSection \\ref{section:model} establishes model identifiability and introduces the concepts of nondegeneracy and regularity.\nSection \\ref{section:method} develops the proposed methods for structure learning and statistical inference.\nSection \\ref{section:theory} presents statistical theory to justify the proposed methods.\nSection \\ref{section:simulation} performs simulation studies, followed by an application to infer gene pathways with gene expression and SNP data in Section \\ref{section:real-data}. Section \\ref{section:conclusion} concludes the article. \nThe Appendix contains illustrative examples and technical proofs.}\n\n\\section{Gaussian directed acyclic graph with additive interventions}\\label{section:model}\n\nTo infer directed relations among primary variables (variables of primary interest) $\\bm Y =\n(Y_1,\\ldots,Y_p)^\\top$, \nconsider a structural equation model with additive interventions: \n\\begin{eqnarray}\n \\label{equation:model}\n \\bm Y= \\bm U^{\\top} \\bm Y + \\bm W^{\\top} \\bm X + \\bm \\varepsilon, \n \\quad \\bm \\varepsilon \\sim N(\\bm 0, \\bm\\Sigma), \\quad \\bm \\Sigma = \\text{Diag}(\\sigma^2_1,\\ldots,\\sigma^2_p),\n\\end{eqnarray}\nwhere $\\bm X=(X_1,\\ldots, X_q)^\\top$ is a vector of additive intervention variables,\n$\\bm U\\in \\mathbb R^{p \\times p}$ and $\\bm W \\in \\mathbb R^{q \\times p}$ are unknown coefficient matrices, \nand $\\bm \\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_p)^\\top$ is a vector of random errors with $\\sigma_j^2>0$; $j=1,\\ldots,p$. In \\eqref{equation:model}, $\\bm \\varepsilon$ is independent\nof $\\bm X$ but components of $\\bm X$ can be dependent.\nThe matrix $\\bm U$ specifies the directed relations among $\\bm Y$, \nwhere $U_{kj}\\neq 0$ if $Y_k$ is a direct cause of $Y_j$, denoted by $Y_k \\rightarrow Y_j$, {and $Y_k$ is called a {parent} of $Y_j$ or $Y_j$ a {child} \nof $Y_k$.}\nThus, $\\bm U$ represents a directed graph, which is required to be acyclic to ensure the validity of the local Markov property \\citep{spirtes2000causation}.\nThe matrix $\\bm W$ specifies the targets and strengths of interventions, \nwhere $W_{lj}\\neq 0$ indicates $X_l$ intervenes on $Y_j$, denoted by \n$X_l\\to Y_j$. In \\eqref{equation:model}, no directed edge from a primary \nvariable $Y_j$ to an intervention variable $X_l$ is permissible. \n\n{In what follows, we will focus on the DAG $G$ with nodes $\\bm Y=(Y_1,\\ldots,Y_p)$, primary variable edges $\\mathcal E = \\{ (k,j) : U_{kj}\\neq 0 \\}$, and intervention edges $\\mathcal I=\\{ (l,j) : W_{lj}\\neq 0 \\}$.}\n\n\n\\subsection{Identifiability}\\label{section:identifiability}\n\n{\nModel \\eqref{equation:model} is generally non-identifiable without interventions \n($\\bm W = \\bm 0$) when errors do not meet\nsome requirements such as the equal-variance assumption \\citep{bulhmann2013inde} and its variants\n\\citep{ghoshal2018learning,rajendran2021structure}.}\nMoreover, a model can be identified when $\\bm \\varepsilon$ in \\eqref{equation:model}\nis replaced by non-Gaussian errors \\citep{shimizu2006linear}\nor linear relations are replaced by nonlinear ones \n\\citep{hoyer2009nonlinear}. Regardless, suitable interventions can make \n\\eqref{equation:model} identifiable. \nWhen intervention targets are known, \nthe identifiability issue has been studied \n\\citep{oates2016estimating,chen2018two}. \nHowever, it is less so when the exact targets and strengths of interventions are unknown,\nas in many biological applications \\citep{jackson2003expression,kulkarni2006evidence}, \nreferring to the case of \\emph{unspecified} or \\emph{uncertain} interventions \n\\citep{heinze2018causal,eaton2007exact,squires2020permutation}. \n\n\nWe now categorize interventions as instruments and invalid instruments.\n\n\\begin{definition}[Instrument] \\label{definition:instrument}\nAn intervention is an instrument in a DAG if it satisfies:\n\\begin{enumerate}\n \\item [(A)] (Relevance) it intervenes on at least one primary variable; \n \\item [(B)] (Exclusion) it does not intervene on more than one primary variable.\n\\end{enumerate}\nOtherwise, it is an invalid instrument.\n\\end{definition}\n\nHere, (A) requires an intervention \nto be active, while (B) prevents simultaneous interventions of a single \nintervention variable on multiple primary variables. This is critical to identifiability\nbecause in this situation an instrument on a cause variable $Y_1$ helps reveal its directed effect on \nan outcome variable $Y_2$, which would otherwise be impossible. \nThus, the instruments are introduced to break the symmetry that results in non-identifiability of directed relations $Y_1 \\rightarrow Y_2$ and $Y_2 \\rightarrow Y_1$.\n\n\nNext, we make some assumptions on intervention variables to yield an identifiable \nmodel, where dependencies among intervention variables are permissible.\n \n\\begin{assumption}\\label{assumption:identifiability}\n Assume that model \\eqref{equation:model} satisfies the following conditions.\n \\begin{enumerate}\n \\item [(1A)] (Positive definiteness) $\\operatorname{\\mathbb E}\\bm X\\bm X^\\top$ is positive definite, where $\\operatorname{\\mathbb E}$ is the expectation.\n \n \\item [(1B)] (Local faithfulness) $\\operatorname{Cov}(Y_j, X_l \\mid \\bm X_{\\{1,\\ldots,q\\}\\setminus\\{l\\}})\\neq 0$ when $X_l$ intervenes on $Y_j$, where $\\operatorname{Cov}$ denotes the covariance.\n\n \\item [(1C)] (Instrumental sufficiency) Each primary variable is intervened by at least one instrument.\n \\end{enumerate}\n\\end{assumption}\n\nAssumption 1A imposes mild distributional restrictions on $\\bm X$, permitting discrete variables such as SNPs. Assumption 1B requires interventional effects not to cancel out as multiple targets from an invalid instrument are permitted. \nImportantly, if either Assumption 1B or Assumption 1C fails, model \\eqref{equation:model} is generally not identifiable, as shown in Example \\ref{example:identifiability} of Appendix \\ref{section:identifiability-example}. \nIn Section \\ref{section:simulation}, we empirically examine the situation when Assumption 1C is not met.\n\n\n\\begin{proposition}[Identifiability]\\label{proposition:identifiability}\nUnder Assumption \\ref{assumption:identifiability}, {the parameters $(\\bm U, \\bm W, \\bm\\Sigma)$ in model \\eqref{equation:model} are identifiable.}\n\\end{proposition}\n\nProposition \\ref{proposition:identifiability} {(proved in Appendix \\ref{proof:identifiability})} is derived for a DAG model with unspecified interventions. \nThis is in contrast to Proposition 1 of \\citet{chen2018two}, which proves the identifiability of the parameters in a directed graph with target-known instruments on each primary variable.\nMoreover, it is worth noting that the estimated graph in \\citet{chen2018two} may be cyclic and lacks the local Markov property for causal interpretation \\citep{spirtes2000causation}. \n\n\\subsection{Likelihood-based inference for a DAG}\\label{section:testability}\n\nOur primary goal is to perform statistical inference of directed edges and pathways. \nLet $\\mathcal{H}\\subseteq \\{ (k,j):k\\neq j, 1\\leq k,j\\leq p \\}$ be an edge set among primary variables $\\{ Y_1,\\ldots,Y_p \\}$, where $(k,j) \\in \\mathcal{H}$ \nspecifies a (hypothesized) directed edge $Y_k \\to Y_j$ in \\eqref{equation:model}. \nWe shall focus on two types of testing with null and alternative hypotheses $H_0$ and $H_a$.\nFor simultaneous testing of directed edges,\n\\begin{eqnarray}\n\\label{equation:link-test}\nH_0: U_{kj}= 0; \\text{ for all } (k,j) \\in \\mathcal{H} \\quad \\mbox{ versus } \n\\quad H_a: U_{kj}\\neq 0 \\text{ for some } (k,j) \\in \\mathcal{H}; \n\\end{eqnarray}\nfor simultaneous testing of pathways, \n\\begin{eqnarray}\n\\label{equation:pathway-test}\nH_0: U_{kj}= 0; \\text{ for some } (k,j) \\in \\mathcal{H} \\quad \\mbox{ versus }\n\\quad H_a: U_{kj} \\neq 0 \\text{ for all } (k,j) \\in \\mathcal{H},\n\\end{eqnarray}\nwhere $(\\bm U_{\\mathcal {H}^c},\\bm W, \\bm \\Sigma)$ \nare unspecified nuisance parameters and $^c$ is the set complement. \n\n{Note that $H_0$ in \\eqref{equation:pathway-test} is a composite hypothesis that can be decomposed into sub-hypotheses\n \\begin{equation*}\n H_{0,\\nu}: U_{k_\\nu,j_\\nu}= 0, \\quad \\mbox{ versus } \n \\quad {H}_{a,\\nu}: U_{k_\\nu,j_\\nu}\\neq 0; \\quad \\nu=1,\\ldots,|\\mathcal H|,\n \\end{equation*}\nwhere $\\mathcal{H}=\\{ (k_1,j_1),\\ldots,(k_{|\\mathcal H|},j_{|\\mathcal H|}) \\}$ and each sub-hypothesis is a directed edge test.}\n\nNow consider likelihood inference for \\eqref{equation:link-test}.\nGiven an independent and identically distributed sample $(\\bm Y_i,\\bm X_i)_{i=1}^n$ from \n\\eqref{equation:model}, the log-likelihood is \n\\begin{equation}\n\\label{equation:likelihood}\n L(\\bm\\theta,\\bm\\Sigma)= -\n \\frac{1}{2} \\sum_{i=1}^n\n \\|\\bm\\Sigma^{-1\/2} \n ((\\bm I - \\bm U^\\top)\\bm Y_i - \\bm W^\\top \\bm X_i ) \\|^2_2 - \n n\\log\\sqrt{\\det(\\bm\\Sigma)},\n\\end{equation}\nwhere $\\bm\\theta = (\\bm U,\\bm W)$, $\\bm \\Sigma=\\text{Diag}(\\sigma_1^2,\\ldots,\\sigma_p^2)$, and $\\bm U$ is subject to the acyclicity constraint \\citep{zheng2018dags,yuan2019constrained} that no directed cycle is permissible in\na DAG. The likelihood ratio is defined as \n$\\text{Lr} = L(\\widehat{\\bm\\theta}^{(1)},\\widehat{\\bm\\Sigma}) - L(\\widehat{\\bm\\theta}^{(0)},\\widehat{\\bm\\Sigma})$,\nwhere $\\widehat{\\bm\\theta}^{(1)}$ is a maximum likelihood estimate (MLE) in \\eqref{equation:likelihood} subject to the acyclicity constraint,\n$\\widehat{\\bm\\theta}^{(0)}$ is an MLE subject to an additional requirement $\\bm U_{\\mathcal H} = \\bm 0$,\nand $\\widehat{\\bm\\Sigma}$ is an estimator of $\\bm\\Sigma$.\n\nIn many statistical models, {a likelihood ratio has a nondegenerate and tractable distribution}, for instance, a chi-squared distribution with degrees of freedom $|\\mathcal{H}|$.\nHowever, in \\eqref{equation:model}, because of the acyclicity constraint, {some hypothesized edges in $\\mathcal H$ can be absent given the edge subset $\\mathcal E\\cap \\mathcal H^c$ defined by the nonzero elements of nuisance parameter $\\bm U_{\\mathcal H^c}$, where $\\mathcal{E}$ denotes the edge set of the whole DAG.} \nAs a result, $\\text{Lr}$ may converge to a distribution with degrees of freedom less than $|\\mathcal H|$ and the distribution may be even intractable, making inference for a DAG greatly different from the classical ones, as illustrated by Example \\ref{example:testability}.\n\n\\begin{figure}\n \\centering\n \n \\includegraphics[width=0.45\\textwidth]{example1a.pdf}\n \n \n \\caption{A true DAG structure of five primary variables \n$Y_1,\\ldots,Y_5$ and five intervention variables $X_1,\\ldots,X_5$, where\ndirected edges are represented by solid arrows while \ndependencies among $\\bm X$ are not displayed.} \n\\label{figure:testability}\n\\end{figure}\n\n\\begin{example}\\label{example:testability} Consider the likelihood ratio test under null $H_0$ \nand alternative $H_a$ for the DAG displayed in Figure \\ref{figure:testability}. {Let $\\mathcal E$ be the edge set of the DAG.}\n\\begin{itemize}\n \\item $H_0: U_{21} = 0$ versus $H_a: U_{21}\\neq 0$, where $\\mathcal H=\\{ (2,1) \\}$. {Here, $(2,1)$ forms a cycle together with the edges in $\\mathcal E \\cap \\mathcal H^c$ (namely the edges not considered by the hypothesis), violating the acyclicity constraint.} Note that the maximum likelihood value $L(\\widehat{\\bm\\theta},\\widehat{\\bm\\Sigma})$ with $U_{21}\\neq 0$ tends to be smaller than that with $U_{21}=0$ when a random sample is obtained under $H_0$, especially so when the asymptotics kicks in as the sample size increases. Consequently, the likelihood ratio $\\text{Lr}$ becomes zero, constituting a degenerate situation.\n\n \\item $H_0: U_{45} = U_{53} = 0$ versus $H_a: U_{45}\\neq 0$ or $U_{53}\\neq 0$, where $\\mathcal H=\\{ (4,5),(5,3)\\}$.\n {In this case, $\\{(4,5),(5,3)\\}$ forms a cycle with the edges in $\\mathcal E\\cap\\mathcal H^c$, violating the acyclicity constraint.} \n Since the likelihood value $L({\\bm\\theta},\\bm\\Sigma)$ tends to be maximized under the true graph when data is sampled under $H_0$, we have $L(\\widehat{\\bm\\theta}^{(1)},\\widehat{\\bm\\Sigma}) = \\max(L(\\widehat{U}_{45},\\widehat{U}_{53}=0,\\widehat{\\bm U}_{\\mathcal H^c},\\widehat{\\bm W},\\widehat{\\bm\\Sigma}), L(\\widehat{U}_{45}=0,\\widehat{U}_{53},\\widehat{\\bm U}_{\\mathcal H^c},\\widehat{\\bm W},\\widehat{\\bm\\Sigma}))$. As a result, the likelihood ratio distribution becomes complicated in this situation due to the dependence between the two components in $L(\\widehat{\\bm\\theta}^{(1)},\\widehat{\\bm\\Sigma})$.\n\\end{itemize}\n\\end{example}\n\n\n\nMotivated by Example \\ref{example:testability}, we introduce the concepts of nondegeneracy and regularity.\n\n\\begin{definition}[Nondegeneracy and regularity with respect to DAG]\\label{definition:testability} ~\n\\begin{enumerate}[(A)]\n \\item An edge $(k,j)\\in \\mathcal{H}$ is nondegenerate with respect to DAG $G$ if $\\{(k,j)\\}\\cup \\mathcal{E}$ contains no directed cycle,\n where $\\mathcal{E}$ denotes the edge set of $G$. Otherwise, $(k,j)$ is degenerate.\n Let $\\mathcal{D} \\subseteq \\mathcal{H}$ be the set of all nondegenerate edges with respect to $G$.\n A null hypothesis $H_0$ is nondegenerate with respect to DAG $G$ if $\\mathcal{D}\\neq\\emptyset$.\n Otherwise, $H_0$ is degenerate.\n \\item A null hypothesis $H_0$ is said to be regular with respect to DAG $G$ if $\\mathcal{D} \\cup \\mathcal{E}$ contains no directed cycle, where $\\mathcal{E}$ denotes the edge set of $G$. Otherwise, $H_0$ is called irregular.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\n In practice, $\\mathcal{D}$ is unknown and needs to be estimated from data.\n\\end{remark}\n\nNondegeneracy ensures nonnegativity of the likelihood ratio. \n{In testing \\eqref{equation:link-test}, \nregularity excludes intractable situations for the null distribution.\nIn testing \\eqref{equation:pathway-test}, if $H_0$ is irregular, \nthen $\\mathcal D\\cup\\mathcal{E}$ has a directed cycle,\nwhich means the hypothesized directed pathway cannot exist due to the acyclicity constraint. \nThus, regularity excludes the degenerate situations in testing \\eqref{equation:pathway-test}.}\n\nIn what follows, we mainly focus on nondegenerate and regular hypotheses. \nFor the degenerate case, the p-value is defined to be one.\n{For the irregular case of edge test \\eqref{equation:link-test}, \nwe decompose the hypothesis into regular sub-hypotheses and conduct multiple testing.\nFor the irregular case of pathway test \\eqref{equation:pathway-test}, the p-value is defined to be one.\nMore discussions on the implementation in irregular cases are provided in Appendix \\ref{section:appendix-irregular-hypothesis}.}\n\n\n\nFinally, we introduce some notations for a DAG $G$ with primary \nvariables $\\{Y_1,\\ldots,Y_p\\}$, intervention variables $\\{ X_1,\\ldots,X_q \\}$, a directed edge set $\\mathcal E\\subseteq \n\\{(k,j):k\\neq j, 1\\leq k,j\\leq p \\}$, and an intervention edge set $\\mathcal I \\subseteq \\{ (l,j) : 1\\leq l\\leq q, 1\\leq j\\leq p \\}$. \nIf there is a directed path $Y_k\\to\\cdots\\to Y_j$ in $G$, \n$Y_k$ is called an ancestor of $Y_j$ or $Y_j$ a descendant of $Y_k$.\nLet $\\mathcal A = \\{ (k,j) : Y_k\\to\\cdots\\to Y_j\\}$ be the set of all ancestral relations. \nIn a DAG $G$, let $\\textnormal{\\textsc{pa}}_G(j) = \\{ k : (k,j)\\in\\mathcal E \\}$, \n$\\textnormal{\\textsc{an}}_G(j) = \\{ k: (k,j)\\in \\mathcal A \\}$, $\\textnormal{\\textsc{d}}_G(j) = \\{ k: (k,j) \\in\\mathcal D \\}$, and $\\textnormal{\\textsc{in}}_G(j)=\\{ l : (l,j)\\in\\mathcal I \\}$ be the parent, ancestor, nondegenerate hypothesized parent, and intervention sets of $Y_j$, respectively.\n\n\n\n\\section{Methodology}\\label{section:method}\n\nThis section develops the main methodology, \nincluding the peeling algorithm for structure learning and the data perturbation inference for \nsimultaneous testing of directed edges \\eqref{equation:link-test} and pathways \\eqref{equation:pathway-test}.\n\nOne major challenge to the likelihood approach is optimization subject to the acyclicity requirement. This constraint imposes difficulty on not only computation but also asymptotic theory. As a result, there is a gap between the asymptotic distribution based on a global maximum and that on the actual estimate which could be a local maximum \\citep{jankova2018inference,li2019likelihood}. Moreover, the actual estimate may give an imprecise topological order, tending to impact adversely on inference.\n\nTo circumvent the acyclicity requirement, we propose using a super-graph $S$ of the true DAG $G$, where $S$ is constructed by a novel peeling method without explicitly imposing the acyclicity constraint. As a result, our method is scalable with a statistical guarantee of the actual estimate.\n{We will focus on a specific super-graph $S$ consisting of primary variable edges $\\mathcal E_{S} := \\mathcal A\\supseteq \\mathcal E$} and candidate intervention edges $\\mathcal I_{S} := \\mathcal C \\supseteq \\mathcal I$, \nwhere $\\mathcal{E}$ and $\\mathcal I$ are sets of primary variable edges and intervention edges in the original graph $G$, {and set $\\mathcal{C}$ is to be defined shortly after \\eqref{equation:model2}}. Note that $S$, whose primary variable edges consist of all ancestral relations of $G$, is also a DAG. \nGiven the super-graph $S$, the true DAG $G$ can be reconstructed by nodewise constrained regressions.\n\nLet $\\bm \\theta_{S}=\\big(\\bm U_{\\mathcal A}, \\bm W_{\\mathcal C}\n\\big)$ be $\\bm \\theta$ restricted to $S$. Note that \n$\\bm\\theta_{S^c} = (\\bm U_{\\mathcal A^c},\\bm W_{\\mathcal C^c}) = \\bm 0$ in the original graph $G$.\nGiven $S$, the log-likelihood $L(\\bm \\theta,\\bm\\Sigma)$ in \\eqref{equation:likelihood} can be rewritten as\n\\begin{equation}\n\\label{equation:likelihood-supergraph}\n \\begin{split}\n L_S(\\bm \\theta_S,\\bm \\Sigma)\n &= -\\sum_{j = 1}^p \\Big(\\text{RSS}_j(\\bm \\theta_{S})\/2\\sigma_j^2 + \\log\\sqrt{2\\pi\\sigma_j^2}\\Big),\\\\\n \\text{RSS}_j(\\bm\\theta_{S})\n &= \\sum_{i=1}^n \\Big(Y_{ij} - \\sum_{(k,j)\\in\\mathcal A} U_{kj}Y_{ik} - \\sum_{(l,j)\\in\\mathcal C} W_{lj}X_{il} \\Big)^2,\\\\\n \\end{split}\n\\end{equation} \nwhich involves $|\\mathcal{A}|+|\\mathcal C|$ parameters for $\\bm \\theta_{S}$.\n\nOur plan is as follows. \nIn Section \\ref{section:causal-discovery}, we hierarchically construct ${S}$ via nodewise regressions of $Y_j$ over $\\bm X$; $j=1,\\ldots,p$ without the acyclicity constraint for $\\bm U$.\nThen in Section \\ref{section:dp-inference}, on the ground of constructed super-graph $S$ and \\eqref{equation:likelihood-supergraph}, we develop likelihood ratio tests for hypotheses \\eqref{equation:link-test} and \\eqref{equation:pathway-test}. \n\n\n\\subsection{Structure learning via peeling}\\label{section:causal-discovery}\n\nThis section develops a novel structure learning method based on a\npeeling algorithm and nodewise constrained regressions to construct \n$S=(\\mathcal{A},\\mathcal C)$ in a hierarchical manner.\n\nFirst, we observe an important connection between primary variables and \nintervention variables. \nRewrite \\eqref{equation:model} as\n\\begin{equation}\n\\label{equation:model2}\n\\bm Y = \\bm V^\\top \\bm X + \\bm \\varepsilon_V, \\quad \\bm \\varepsilon_V=(\\bm I-\\bm U^\\top)^{-1}\\bm \\varepsilon \\sim N(\\bm 0,\\bm\\Omega^{-1}),\n\\end{equation}\nwhere $\\bm\\Omega= (\\bm I - \\bm U)\\bm\\Sigma^{-1}(\\bm I -\\bm U^\\top)$ is a precision matrix \nand $\\bm V = \\bm W (\\bm I - \\bm U)^{-1}$. \nNow, define the set of candidate interventional relations $\\mathcal C := \\{ (l,j) : V_{lj}\\neq 0 \\}$.\n\n\n\\begin{proposition} \\label{proposition:peeling} \n Suppose Assumption \\ref{assumption:identifiability} is satisfied. Then $\\mathcal C\\supseteq \\mathcal I$. Moreover,\n \\begin{enumerate}[(A)]\n \\item if $V_{lj}\\neq 0$, then $X_{l}$ intervenes on $Y_j$ or an ancestor of $Y_j$; \n \\item $Y_j$ is a leaf variable (having no child)\n if and only if there is an instrument $X_l$ such that $V_{lj}\\neq 0$ and $V_{lj'}=0$ for $j'\\neq j$. \n \\end{enumerate}\n\\end{proposition}\n\n{The proof of Proposition \\ref{proposition:peeling} is deferred to Appendix \\ref{proof:peeling}.}\nIntuitively, $V_{lj}\\neq 0$ implies the dependence of $Y_j$ on $X_{l}$ through a directed path $X_l\\to \\cdots \\to Y_j$, and \nhence that $X_l$ intervenes on $Y_j$ or an ancestor of $Y_j$. \nThus, the instruments on a leaf variable are independent of the other primary variables conditional on the rest of interventions. \nThis observation suggests a method to reconstruct the DAG topological order by sequentially identifying the leaves in the DAG and removing the identified leaf variables. \n\nNext, we discuss the estimation of $\\bm V$ and construction of $S$.\n\n\\subsubsection{Nodewise constrained regressions}\n\nWe propose nodewise $\\ell_0$-constrained regressions to estimate\nnonzero elements of $\\bm V$:\n\\begin{equation}\n \\label{pseudo-likelihood}\n \\widehat{\\bm V}_{\\cdot j} = \\mathop{\\rm arg\\, min\\,}_{\\bm V_{\\cdot j}} \\ \n \\sum_{i=1}^n (Y_{ij} - \\bm V_{\\cdot j}^\\top \\bm X_i)^2\n \\quad \\text{s.t.} \\quad \\sum_{l=1}^q \\operatorname{I}(V_{lj}\\neq 0) \\leq \\kappa_j; \\quad j=1,\\ldots,p,\n\\end{equation}\nwhere $1 \\leq \\kappa_j\\leq q$ is an integer-valued tuning parameter controlling the sparsity of $\\bm V_{\\cdot j}$ and can be chosen by BIC or cross-validation.\n\nTo solve \\eqref{pseudo-likelihood}, we use $J_{\\tau_j}(z) = \\min(|z|\/\\tau_j,1)$ as a surrogate of the $\\ell_0$-function $\\operatorname{I}(z \\neq 0)$ \\citep{shen2012likelihood} and {develop a difference-of-convex (DC) algorithm with the $\\ell_0$-projection to improve globality of the solution of \\eqref{pseudo-likelihood}.} \nAt $(t+1)$th iteration, \nwe solve a weighted Lasso problem, \n\\begin{eqnarray}\n \\label{equation:penalized-regression}\n \\widetilde{\\bm V}_{\\cdot j}^{[t+1]}=\\mathop{\\rm arg\\, min\\,}_{\\bm V_{\\cdot j}} \\\n \\sum_{i=1}^n (Y_{ij} -\n \\bm V_{\\cdot j}^\\top \\bm X_i)^2+2n\\gamma_j \\tau_j \\sum_{l=1}^q \\operatorname{I}\\big(|\\widetilde{V}^{[t]}_{lj}|\\leq \\tau_j\\big) \n |V_{lj}|; \\quad j=1,\\ldots,p, \n\\end{eqnarray}\nwhere $\\gamma_j >0$ is a tuning parameter and $\\widetilde{\\bm V}^{[t]}_{\\cdot j}$ is the solution of \\eqref{equation:penalized-regression} at the $t$-th iteration. \nThe DC algorithm terminates at $\\widetilde{\\bm V}_{\\cdot j} = \\widetilde{\\bm V}_{\\cdot j}^{[t]}$ such that $\\|\\widetilde{\\bm V}_{\\cdot j}^{[t+1]}-\\widetilde{\\bm V}_{\\cdot j}^{[t]}\\|_{\\infty}\\leq \\sqrt{\\text{tol}}$, where $\\text{tol}$ is the machine precision.\nThen, we obtain the solution $\\widehat{\\bm V}_{\\cdot j}$ of \\eqref{pseudo-likelihood} by projecting $\\widetilde{\\bm V}_{\\cdot j}$ onto the \n$\\ell_0$-constrained set $\\{\\|\\bm V_{\\cdot j}\\|_0 \\leq \\kappa_j\\}$. \n\n\n\n\\begin{algorithm}\n\nSpecify $\\gamma_j>0$, $\\tau_j>0$, and $1\\leq \\kappa_j\\leq q$.\nInitialize $\\widetilde{\\bm V}_{\\cdot j}^{[0]}$ with $|\\{l:\\widetilde{V}^{[0]}_{l j} >\\tau_j \\}|\\leq \\kappa_j$. \n\nCompute $\\widetilde{\\bm V}_{\\cdot j} = \\widetilde{\\bm V}^{[t]}_{\\cdot j}$ such that \n$\\|\\widetilde{\\bm V}_{\\cdot j}^{[t+1]}-\\widetilde{\\bm V}_{\\cdot j}^{[t]}\\|_{\\infty}\\leq \\sqrt{\\text{tol}}$.\n\n\n{($\\ell_0$-projection)}\nLet $B_j = \\{ l : \\sum_{l'=1}^q \\operatorname{I}( |\\widetilde{V}_{l'j}| \\geq |\\widetilde{V}_{lj}| ) \\leq \\kappa_j \\}$. \nSet \n$$\\widehat{\\bm V}_{\\cdot j}= \\mathop{\\rm arg\\, min\\,}_{\\bm V_{\\cdot j}} \\\n \\|\\bm{\\mathsf{Y}}_{j} - \\bm V_{\\cdot j}^\\top \\bm{\\mathsf{X}}\\|_2^2 \\quad\\text{s.t.}\\quad\nV_{lj}=0 \\text{ for } l \\notin B_j.$$\n\n\\caption{Constrained minimization via DC programming + $\\ell_0$ projection} \\label{algorithm:nodewise-regression}\n\\end{algorithm}\n\n\\begin{algorithm}\n\n \n Let $\\widehat{\\bm V}^{[0]}=\\widehat{\\bm V}$. \n Begin iteration with $h=0,\\ldots$: at iteration $h$, \n \n \n In the current sub-graph identify all instrument-leaf pairs $X_l\\to Y_j$ \n satisfying $l = \\mathop{\\rm arg\\, min\\,}_{l':\\|\\widehat{\\bm V}^{[h]}_{l'\\cdot}\\|_0 \\neq 0} \\|\\widehat{\\bm V}^{[h]}_{l' \\cdot}\\|_0$ and\n $j=\\mathop{\\rm arg\\, max\\,}_{j'} |\\widehat{V}^{[h]}_{l j'}|$. \n Let $h_{\\widehat G}(j)=h$.\n \n \n \n For a leaf $Y_k$, identify all directed relations $Y_k \\to Y_j$ with $h_{\\widehat G}(j)=h-1$ when $\\widehat{V}_{lj}\\neq 0$ for each instrument $X_l$ of $Y_k$ in the current sub-graph. \n \n \n Peeling-off all the leaves $Y_j$\n by deleting $j$th columns from $\\widehat{\\bm V}^{[h]}$ to yield\n $\\widehat{\\bm V}^{[h+1]}$. \n \n Repeat Steps 2-4 until all primary variables are removed. \n \n Compute $\\widehat{S} = (\\widehat{\\mathcal{A}},\\widehat{\\mathcal{C}})$ and $\\widehat{\\mathcal D}$: \n \\begin{equation*}\n \\begin{split}\n \\widehat{\\mathcal{A}}&=\\Big\\{(k,j): Y_k \\to \\cdots \\to Y_j\\Big\\}, \\quad \n \\widehat{\\mathcal D}=\\Big\\{(k,j)\\in \\mathcal{H}: (j,k)\\notin \\widehat{\\mathcal A} \\Big\\},\\\\\n \\widehat{\\mathcal{C}}&=\\Big\\{(l,j): |\\widehat{V}_{lj}|> \\tau_j \\Big\\}, \\text{ where $\\tau_j$ is specified in Algorithm \\ref{algorithm:nodewise-regression}.}\n \\end{split}\n \\end{equation*}\n \n \\caption{Reconstruction of topological layers by peeling} \\label{algorithm:peeling}\n \\end{algorithm}\n\n\n\\subsubsection{Peeling}\n\nNow, we describe a \\emph{peeling} algorithm to estimate $S=(\\mathcal{A},\\mathcal C)$ based on $\\bm V$. \n\nBefore proceeding, we introduce the notion of \\emph{topological height} to characterize the DAG topological order. Given a DAG $G$, the height $h_G(j)$ of $Y_j$ is defined as the maximal length of a directed path from $Y_j$ to a leaf variable, where $0\\leq h_G(j) \\leq p-1$ with a leaf having zero height. \nThen the primary variables of the same height form a topological layer and the layers constitute a partition of primary variables in $G$. \nFor instance, in Figure \\ref{figure:testability}, \n$\\{ Y_4,Y_5 \\},\\{ Y_3 \\},\\{ Y_2 \\},\\{Y_1\\}$ are topological layers with height $0,1,2,3$, respectively.\n \n Observe that the topological layer of height $h$ are leaves in the sub-graph with primary variables of height no less than $h$. Based on Proposition \\ref{proposition:peeling}, the peeling \nalgorithm sequentially identifies and peels off the primary variables of height $h$;\n $h=0,1,2,\\ldots$. Note that instruments in the sub-graph may not be necessarily the \ninstruments in the graph $G$ after some primary variables are removed. However, \nthe next proposition indicates that the instruments in the corresponding sub-graph remain \nuseful to reconstruct ancestral relations.\n\n\n\\begin{proposition}\\label{proposition:ancestral-relation}\n Let $Y_k$ have height $h_G(k)$. Let $G'$ be the sub-graph with primary variables of height $\\geq h_G(k)$. Under Assumption \\ref{assumption:identifiability},\nfor a pair $(k,j)$ with $h_G(j) = h_G(k)-1$, we have $Y_k \\to Y_j$ if and only if $V_{lj}\\neq 0$ for \neach instrument $X_l$ of $Y_k$ in $G'$.\n\\end{proposition}\n\n\nProposition \\ref{proposition:ancestral-relation} {(proved in Appendix \\ref{proof:ancestral-relation})} permits the construction of\ndirected relations between the topological layers of heights $h$ and $h-1$; $h\\geq 1$ and thus ancestral relations.\n\n\n\nThe peeling algorithm is summarized in Algorithm \\ref{algorithm:peeling} and \na detailed illustration is presented in Example \\ref{example:peeling} of Appendix \\ref{section:peeling-example}. \n\n\n\n\\begin{remark}[Recovery of true graph $G$]\n$\\widehat{\\mathcal A}$ estimates a superset of directed relations of $\\bm U$ and $\\widehat{\\mathcal C}$ \na superset of interventional relations of $\\bm W$. Therefore, under some technical conditions, with suitable tuning parameters,\nthe true graph $G$ can be reconstructed by estimating $(\\bm U,\\bm W)$ \nthrough $\\ell_0$-constrained regressions given the estimated super-graph\n$\\widehat{S} = (\\widehat{\\mathcal A},\\widehat{\\mathcal{C}})$,\n\\begin{equation}\\label{directed-likelihood}\n \\begin{split}\n \\min_{(\\bm U_{\\textnormal{\\textsc{an}}_{\\widehat{S}}(j), j}, \\bm W_{\\textnormal{\\textsc{in}}_{\\widehat{S}}(j), j})} \n \\sum_{i=1}^n \\left(Y_{ij} -\\bm U_{\\textnormal{\\textsc{an}}_{\\widehat{S}}(j), j}^{\\top} \\bm Y_{i,\\textnormal{\\textsc{an}}_{\\widehat{S}}(j)} - \n\\bm W_{\\textnormal{\\textsc{in}}_{\\widehat S}(j), j}^{\\top} \\bm X_{i,\\textnormal{\\textsc{in}}_{\\widehat S}(j)}\\right)^2\\\\\n \\text{s.t.} \\quad \\sum_{k\\in\\textnormal{\\textsc{an}}_{\\widehat S}(j)} \\operatorname{I}(U_{kj}\\neq 0) \n + \\sum_{l\\in\\textnormal{\\textsc{in}}_{\\widehat S}(j)} \\operatorname{I}(W_{lj}\\neq 0)\n \\leq \\kappa_j'.\n \\end{split}\n\\end{equation}\nThe final estimates $\\widehat{\\bm U}_{\\cdot j} = \n\\Big(\\widehat{\\bm U}_{\\textnormal{\\textsc{an}}_{\\widehat S}(j),j},\\bm 0_{\\textnormal{\\textsc{an}}_{\\widehat S}(j)^c}\\Big)$\nand $\\widehat{\\bm W}_{\\cdot j} = \n\\Big(\\widehat{\\bm W}_{{\\textnormal{\\textsc{in}}}_{\\widehat S}(j),j},\\bm 0_{{\\textnormal{\\textsc{in}}}_{\\widehat S}(j)^c}\\Big)$;\n$j=1,\\ldots,p$.\n\\end{remark}\n\n\n\n\n\\subsection{Likelihood ratio test via data perturbation}\\label{section:dp-inference}\n\n{This subsection proposes an inference method for testing \\eqref{equation:link-test} and \\eqref{equation:pathway-test}.}\nWe first construct a likelihood ratio $\\text{Lr}(S,\\bm \\Sigma)$ in given $(S,\\bm\\Sigma)$ and then plug-in an estimate $(\\widehat{S},\\widehat{\\bm \\Sigma})$.\nNext we perform a test via data perturbation, accounting for the uncertainty of constructing $S$.\n\n\\subsubsection{Likelihood ratio given $S$}\n\nConsider the test in \\eqref{equation:link-test}. From \\eqref{equation:likelihood-supergraph}, the likelihood ratio given $(S,\\bm\\Sigma)$ is\n\\begin{equation*}\n \\text{Lr}(S,\\bm \\Sigma) = L_S(\\widehat{\\bm\\theta}_{S}^{(1)},\\bm \\Sigma) - L_S(\\widehat{\\bm\\theta}_{S}^{(0)}, \\bm \\Sigma)\n= \\sum_{j=1}^p \\frac{\\text{RSS}_j(\\widehat{\\bm\\theta}^{(0)}_{S}) - \\text{RSS}_j(\\widehat{\\bm\\theta}^{(1)}_{S})} {2\\sigma_j^2}\n\\end{equation*}\nwhere $\\widehat{\\bm\\theta}_{S}^{(1)}$ is the MLE given $(S,\\bm\\Sigma)$, and $\\widehat{\\bm\\theta}_{S}^{(0)}$ is \nthe MLE given $(S,\\bm\\Sigma)$ subject to $\\bm U_{\\mathcal H} = 0$. \n\n{Now, recall that $\\textnormal{\\textsc{d}}_G(j) = \\{ k:(k,j)\\in\\mathcal{D} \\}$, where $\\mathcal D$ is the set of nondegenerate edges of $H_0$ with respect to $G$ (Definition \\ref{definition:testability}). \nOf note, $\\textnormal{\\textsc{d}}_S(j) = \\textnormal{\\textsc{d}}_G(j)$ since an edge is nondegenerate with respect to $S$ if and only if it is nondegenerate with respect to $G$.\nTo eliminate the nuisance parameters, observe that if $\\textnormal{\\textsc{d}}_S(j) = \\emptyset$, then \n${\\text{RSS}}_{j}(\\widehat{\\bm\\theta}^{(0)}_{S})-{\\text{RSS}}_{j}(\\widehat{\\bm\\theta}^{(1)}_{S})\n= 0$,\nbecause no $H_0$ constraint is imposed for $Y_{j}$. \nHence, $\\text{Lr}(S,\\bm \\Sigma)$ only summarizes the contributions\nfrom the primary variables with nondegenerate hypothesized edges,\n\\begin{equation}\\label{equation:likelihood-ratio-edge}\n \\text{Lr}(S,\\bm\\Sigma)\n = \\sum_{j : \\textnormal{\\textsc{d}}_S(j)\\neq \\emptyset} \n \\frac{{\\text{RSS}}_j(\\widehat{\\bm\\theta}^{(0)}_{S})\n - {\\text{RSS}}_j(\\widehat{\\bm\\theta}^{(1)}_{S})}{2\\sigma_j^2},\n\\end{equation}\nwhere the nuisance parameters for $Y_j$ with $\\textnormal{\\textsc{d}}_S(j)=\\emptyset$ are eliminated.}\n\nThe likelihood ratio test statistic becomes \n$\\text{Lr}=\\text{Lr}(\\widehat{S},\n\\widehat{\\bm \\Sigma})$.\nHere, $\\widehat{S}$ is constructed by Algorithm \\ref{algorithm:peeling} in Section \\ref{section:causal-discovery} \nand $\\widehat{\\bm \\Sigma}=\\text{Diag}(\\widehat{\\sigma}^2_1,\\ldots,\\widehat{\\sigma}^2_p)$,\n\\begin{eqnarray}\n\\label{degree-freedom}\n\\widehat{\\sigma}^2_j=\n\\frac{\\text{RSS}_j(\\widehat{\\bm\\theta}^{(1)}_{\\widehat{S}})}{n-|{\\textnormal{\\textsc{an}}}_{\\widehat S}(j)|-|{\\textnormal{\\textsc{in}}}_{\\widehat S}(j)|},\n\\quad j=1,\\ldots,p, \n\\end{eqnarray}\nwhere ${\\textnormal{\\textsc{an}}}_{\\widehat{S}}(j)$ and ${\\textnormal{\\textsc{in}}}_{\\widehat S}(j)$\nare the estimated ancestor and candidate intervention sets of $Y_j$.\n\n\\subsubsection{Testing directed edges \\eqref{equation:link-test} via data perturbation}\n\nThe likelihood ratio requires an estimation of $S$, \nwhere we must account for the uncertainty of $\\widehat{S}$ for accurate finite-sample inference.\n\nTo proceed, we consider the test statistic $\\text{Lr}$ based on a ``correct'' super-graph $S\\supseteq S^\\circ$, \nwhere $S=(\\mathcal{A}, \\mathcal{C}) \\supseteq S^\\circ=(\\mathcal{A}^\\circ, \\mathcal{C}^\\circ)$ means that \n$\\mathcal{A}\\supseteq\\mathcal A^\\circ$ and $\\mathcal C\\supseteq\\mathcal C^\\circ$ \nand $^\\circ$ denotes the truth. \nIntuitively, {a ``correct'' super-graph distinguishes descendants and nondescendants and thus can help to infer the true directed relations defined by the local Markov property \\citep{spirtes2000causation} without introducing model errors}, yet may lead to a less powerful test when $S$ is much larger than $S^\\circ$.\nBy comparison, a ``wrong'' graph $S \\not \\supseteq S^\\circ$ provides {an incorrect topological order}, and a test based on a ``wrong'' graph may be biased, \naccompanied by an inflated type-I error. \n\nNow, let $\\bm{\\mathsf{Z}}_{n\\times(p+q)} = (\\bm{\\mathsf X}, \\bm{\\mathsf Y})$ be the data matrix, where \n$\\bm{\\mathsf Y}_{n\\times p}=(\\bm Y_1,\\ldots,\\bm Y_n)^\\top$\nand $\\bm{\\mathsf X}_{n\\times q} = (\\bm X_1,\\ldots,\\bm X_n)^\\top$.\nMoreover, let $\\bm{\\mathsf{e}}_j \\in \\mathbb R^n$ be the $j$th column of error matrix\n$\\bm{\\mathsf{e}}_{n\\times p} = (\\bm\\varepsilon_1,\\ldots,\\bm\\varepsilon_n)^\\top$. \nFrom \\eqref{equation:likelihood-ratio-edge}, assuming $\\widehat{S}\\supseteq S^\\circ$ is a ``correct'' super-graph, \nthe likelihood ratio becomes \n\\begin{equation} \\label{equation:likelihood-ratio}\n \\begin{split}\n \\text{Lr} \n &= \\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq\\emptyset} \n \\frac{\\bm{\\mathsf Y}_j^\\top \n (\\bm{\\mathsf{P}}_{ \\widehat A_j } - \\bm{\\mathsf{P}}_{ \\widehat B_j}) \n \\bm{\\mathsf Y}_j }{ 2\\widehat{\\sigma}_j \n } \\\\ \n &= \\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq\\emptyset}\n \\frac{( \\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} + \\bm{\\mathsf e}_j)^\\top (\\bm{\\mathsf{P}}_{ \\widehat A_j } - \\bm{\\mathsf{P}}_{ \\widehat B_j})\n (\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} + \\bm{\\mathsf e}_j)}{2\\widehat{\\sigma}_j}, \\\\\n \\widehat{\\sigma}_j &= \\frac{\\bm{\\mathsf Y}_j^\\top(\\bm I-\\bm{\\mathsf{P}}_{ \\widehat A_j })\\bm{\\mathsf Y}_j}{n-|\\widehat A_j|}=\\frac{\\bm{\\mathsf e}_j^\\top(\\bm I-\\bm{\\mathsf{P}}_{ \\widehat A_j })\\bm{\\mathsf e}_j}{n-|\\widehat A_j|}; \\quad 1\\leq j \\leq p, \n \\end{split}\n\\end{equation}\nwhere $\\bm{\\mathsf{P}}_A = \\bm{\\mathsf Z}_A(\\bm{\\mathsf Z}_A^\\top\\bm{\\mathsf Z}_A)^{-1}\\bm{\\mathsf Z}_A^\\top$ is the projection matrix onto the span of columns $A$ in $\\bm{\\mathsf Z}$,\n$\\widehat{A}_j = {\\textnormal{\\textsc{an}}}_{\\widehat{S}}(j)\\cup {\\textnormal{\\textsc{in}}}_{\\widehat S}(j)\\cup{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)$, and \n{$\\widehat{B}_j = ({\\textnormal{\\textsc{an}}}_{\\widehat{S}}(j)\\cup {\\textnormal{\\textsc{in}}}_{\\widehat S}(j))\\setminus{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)$}.\nOf note, $\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_S(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_S(j),j} = \\bm 0$ for all $j$ under $H_0$,\nwhile $\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_S(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_S(j),j} \\neq \\bm 0$ for some $j$ under $H_a$. \n\nWe propose a data perturbation (DP) method \\citep{shen2002adaptive,breiman1992little} to approximate the null distribution of \\text{Lr} in \\eqref{equation:likelihood-ratio}. The idea behind DP is to assess the sensitivity of the estimates through perturbed data $\\bm{Y}^*_i = \\bm{Y}_i + \\bm \\varepsilon^*_i$, where $\\bm \\varepsilon_i^* \\sim N(0,\\widehat{\\bm\\Sigma})$; $i=1,\\ldots,n$. \n\nLet $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*) = (\\bm{\\mathsf X},\\bm{\\mathsf Y}^*,\\bm{\\mathsf e}^*)$ be\nperturbed data, where $\\bm{\\mathsf e}^*_{n\\times p}=(\\bm{\\mathsf e}^*_1,\\ldots,\\bm{\\mathsf e}^*_p)=(\\bm\\varepsilon^*_1,\\ldots,\\bm\\varepsilon^*_n)^\\top$ denotes perturbation errors. \nGiven $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$, we obtain a perturbation estimate $\\widehat{S}^*$. \nIn \\eqref{equation:likelihood-ratio}, under $H_0$, $\\text{Lr} = f(\\bm{\\mathsf X}, \\bm \\mu, \\bm{\\mathsf e})$ is\na function of intervention data $\\bm{\\mathsf X}$, conditional mean \n$\\bm\\mu = \\operatorname{\\mathbb E}(\\bm{\\mathsf{Y}} \\mid \\bm{\\mathsf{X}}) \n\\in \\mathbb R^{n\\times p}$ and unobserved error $\\bm{\\mathsf e}$.\nHowever, the perturbation error $\\bm{\\mathsf e}^*$ is \\emph{known} given the DP sample \n$(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$, suggesting a form $\\text{Lr}^*$ based on\n$\\text{Lr}$: \n\\begin{equation}\\label{equation:likelihood-ratio-dp}\n \\begin{split}\n \\text{Lr}^* &\\equiv \n f(\\bm{\\mathsf X}, \\bm{\\mathsf Y},\\bm{\\mathsf e}^*)= \n \\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq\\emptyset} \n \\frac{(\\bm{\\mathsf e}_j^*)^\\top (\\bm{\\mathsf{P}}_{\\widehat A^*_j} - \\bm{\\mathsf{P}}_{\\widehat B^*_j})\n \\bm{\\mathsf e}^*_j}{2\\widetilde{\\sigma}_j^*}, \\quad \n \\widetilde{\\sigma}^*_j = \\frac{(\\bm{\\mathsf e}_j^*)^\\top(\\bm I-\\bm{\\mathsf{P}}_{\\widehat A^*_j})\\bm{\\mathsf e}^*_j}{n-|\\widehat A^*_j|}, \\quad 1\\leq j\\leq p,\n \\end{split}\n\\end{equation}\nwhich mimics \\eqref{equation:likelihood-ratio} when $\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_S(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_S(j)} = \\bm 0$ under $H_0$,\nwith $\\bm{\\mathsf e}_j$ replaced by $\\bm{\\mathsf e}^*_j$. \nAs a result, when $\\widehat{S}^*\\supseteq S^\\circ$, the conditional distribution of \n$\\text{Lr}^*$ given $\\bm{\\mathsf Z}$ well approximates\nthe null distribution of $\\text{Lr}$, \nwhere the model selection effect is accounted for by assessing the variability of $\\widehat A^*_j$ and $\\widehat{B}^*_j$ over different realizations of $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$.\n\nIn practice, we use Monte-Carlo to approximate the distribution\nof $\\text{Lr}^*$ given $\\bm{\\mathsf Z}$ in \\eqref{equation:likelihood-ratio-dp}. In particular, we generate $M$ perturbed \nsamples $(\\bm{\\mathsf{Z}}^*_m,\\bm{\\mathsf e}_m^*)_{m=1}^M$ independently and compute $\\text{Lr}^*_m$; $m=1,\\ldots,M$.\nThen, we examine the condition $\\widehat{S}^*_m\\supseteq S^\\circ$ by checking its empirical counterpart \n$\\widehat{S}^*_m\\supseteq \\widehat{S}$.\nThe DP p-value of edge test in \\eqref{equation:link-test} is defined as \n\\begin{equation}\\label{pval:link}\n \\text{Pval} = \\left(\\sum_{m=1}^{M} \n \\operatorname{I}(\\text{Lr}_m^*\\geq \\text{Lr},\\widehat{S}^*_m\\supseteq\\widehat{S})\\right)\/\n \\left( \\sum_{m=1}^{M} \\operatorname{I}(\\widehat{S}^*_m\\supseteq \\widehat{S})\\right),\n\\end{equation}\nwhere $\\operatorname{I}(\\cdot)$ is the indicator function.\n\n\n\n\\begin{remark}\n Assume \n$\\widehat{S}^*\\supseteq S^\\circ$ is a ``correct'' super-graph. \nA naive likelihood ratio based on \nperturbed data $(\\bm{\\mathsf Z}^*,\\bm{\\mathsf e}^*)$ is \n\\begin{equation*}\n \\text{Lr}(\\widehat{S}^*,\\widehat{\\bm\\Sigma}^*)=\\sum_{j:{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)\\neq \\emptyset} \\frac{(\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} + \\bm{\\mathsf e}_j + \\bm{\\mathsf e}^*_j)^\\top\n (\\bm{\\mathsf{P}}_{\\widehat{A}^*_j} - \\bm{\\mathsf{P}}_{\\widehat{B}^*_j})(\\bm{\\mathsf Y}_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j)}\\bm U_{{\\textnormal{\\textsc{d}}}_{\\widehat{S}}(j),j} + \\bm{\\mathsf e}_j + \\bm{\\mathsf e}^*_j) }{ 2 \\widehat{\\sigma}_j^*}.\n\\end{equation*} \nNote that the error $\\bm{\\mathsf e}_j$ given $\\bm{\\mathsf Z}$ is deterministic and does not vanish under either $H_0$ or $H_a$; $j=1,\\ldots,p$. \nThus, the conditional distribution of $\\text{Lr}(\\widehat{S}^*,\\widehat{\\bm\\Sigma}^*)$ given $\\bm{\\mathsf Z}$ does not approximate the \nnull distribution of $\\text{Lr}$ in \\eqref{equation:likelihood-ratio},\nin contrast to the DP likelihood ratio $\\text{Lr}^*$ in \\eqref{equation:likelihood-ratio-dp}.\n\\end{remark}\n\n\n\\subsubsection{Extension to hypothesis testing for a pathway}\n\nNext, we extend the DP inference for \\eqref{equation:pathway-test}. \nDenote $\\mathcal{H}=\\{ (k_1,j_1),\\ldots,(k_{|\\mathcal{H}|},j_{|\\mathcal{H}|}) \\}$. \nThen the test of pathways in \\eqref{equation:pathway-test} can be reduced to testing sub-hypotheses\n\\begin{equation*}\n H_{0,\\nu}: U_{k_\\nu,j_\\nu}= 0, \\quad \\mbox{ versus } \n \\quad {H}_{a,\\nu}: U_{k_\\nu,j_\\nu}\\neq 0; \\quad \\nu=1,\\ldots,|\\mathcal H|,\n\\end{equation*}\nwhere each sub-hypothesis is a directed edge test. \nGiven $(S,\\bm\\Sigma)$, the likelihood ratio \nfor $\\text{H}_{0,\\nu}$ is \n$ \\text{Lr}_\\nu(S,\\bm \\Sigma)\n= L_S(\\widehat{\\bm\\theta}_{S}^{(1)},\\bm \\Sigma) \n- L_S(\\widehat{\\bm \\theta}_{S}^{(0,\\nu)},\\bm\\Sigma)$,\nwhere $\\widehat{\\bm\\theta}_{S}^{(0,\\nu)}$ is the MLE under \nthe constraint that $U_{k_\\nu,j_{\\nu}}=0$. \nThen for $\\widehat{S}\\supseteq S^\\circ$, \nwe have \n\\begin{equation}\n \\begin{split}\n \\text{Lr}_\\nu &= \\frac{1}{2}\n \\frac{( \\bm{\\mathsf Y}_{k_\\nu} U_{k_\\nu, j_{\\nu}} + \\bm{\\mathsf e}_{j_\\nu})^\\top (\\bm{\\mathsf{P}}_{\\widehat A_{j_\\nu}} - \\bm{\\mathsf{P}}_{\\widehat B_{j_\\nu}})\n (\\bm{\\mathsf Y}_{k_\\nu} U_{k_\\nu, j_\\nu} + \\bm{\\mathsf e}_{j_\\nu})}{(\\bm{\\mathsf e}_{j_\\nu})^\\top(\\bm I-\\bm{\\mathsf{P}}_{\\widehat A_{j_\\nu}})\\bm{\\mathsf e}_{j_\\nu}\/(n-|\\widehat A_{j_\\nu}|)}, \n \\\\\n \\text{Lr}_\\nu^* &=\n \\frac{1}{2} \n \\frac{(\\bm{\\mathsf e}^*_{j_\\nu})^\\top (\\bm{\\mathsf{P}}_{\\widehat A^*_{j_\\nu}} - \\bm{\\mathsf{P}}_{\\widehat B^*_{j_\\nu}})\n \\bm{\\mathsf e}^*_{j_\\nu}}{(\\bm{\\mathsf e}^*_{j_\\nu})^\\top(\\bm I-\\bm{\\mathsf{P}}_{\\widehat A^*_{j_\\nu}})\\bm{\\mathsf e}^*_{j_\\nu}\/\n (n-|\\widehat A^*_{j_\\nu}|)},\n \\end{split}\n\\end{equation}\nwhere the distributions of $\\text{Lr}_\\nu^*$ given $\\bm{\\mathsf Z}$\napproximates the null distributions of $\\text{Lr}_\\nu$. \nFinally, define the p-value of pathway test in \\eqref{equation:pathway-test} as\n\\begin{equation}\\label{pval:path}\n \\text{Pval} = \\max_{1\\leq \\nu\\leq |\\mathcal{H}|} \n \\left(\\sum_{m=1}^{M} \\operatorname{I}(\\text{Lr}_{\\nu,m}^*\\geq \\text{Lr}_{\\nu},\\widehat{S}^*_{m}\\supseteq\\widehat{S})\\right)\n \/\\left( \\sum_{i=1}^{M} \\operatorname{I}(\\widehat{S}^*_m\\supseteq \\widehat{S})\\right).\n\\end{equation}\nNote that if any $\\text{H}_{0,\\nu}$ is degenerate, then $\\text{Pval} = 1$.\n\nAlgorithm \\ref{algorithm:data-perturbation} summarizes the DP method for hypothesis testing.\n\n\n\\begin{algorithm}\nSpecify the Monte Carlo size $M$. \n\nCompute Lr and $\\widehat{\\bm \\Sigma}=\\text{Diag}(\\widehat{\\sigma}^2_1,\\ldots,\n\\widehat{\\sigma}^2_p)$\nvia Algorithm \\ref{algorithm:peeling} with original data $\\bm{\\mathsf Z}$.\n\n{(Parallel DP)} \nGenerate perturbed data $(\\bm{\\mathsf Z}^*_m,\\bm{\\mathsf e}_{m}^{*}) = (\\bm{\\mathsf X},\\bm{\\mathsf Y}_{m}^*,\\bm{\\mathsf e}_{m}^{*})$ in parallel, \nwhere $\\bm\\varepsilon_{m,i}^{*}\\sim N(\\bm 0,\\widehat{\\bm\\Sigma})$.\nFor \\eqref{equation:link-test}, compute $\\text{Lr}^*_m$ based on $(\\bm{\\mathsf Z}^*_m,\\bm{\\mathsf e}_{m}^{*})$, while, for \\eqref{equation:pathway-test}, \ncompute $\\text{Lr}^*_{\\nu,m}$; $\\nu=1,\\ldots,|\\mathcal{H}|$ based on $(\\bm{\\mathsf Z}^*_m,\\bm{\\mathsf e}_{m}^{*})$;\n$m=1,\\ldots,M$.\n\nCompute the p-value of a test as \\eqref{pval:link} or \\eqref{pval:path} accordingly.\n \n\\caption{DP likelihood ratio test} \\label{algorithm:data-perturbation}\n\\end{algorithm}\n\n\n\n\\begin{remark}[Computation]\n To accelerate the computation, we consider a parallelizable procedure in Algorithm \\ref{algorithm:data-perturbation}. In addition, we use the original estimate $\\widehat{\\bm\\theta}$ as a warm-start initialization for the DP estimates, which effectively reduces the computing time. \n\\end{remark}\n\n\\begin{remark}[Connection with bootstrap]\n One may consider parametric or nonparametric bootstrap for $\\text{Lr}$. The parametric bootstrap requires a good initial estimate of $(\\bm U,\\bm W)$. Yet, it is rather challenging to correct the bias of this estimate because of the acyclicity constraint. By comparison, DP does not rely on such an estimate. On the other hand, nonparametric bootstrap resamples the original data with replacement. In a bootstrap sample, only about 63\\% distinct observations in the original data are used in model selection and fitting, leading to deteriorating performance \\citep{kleiner2012big}, especially in a small sample. As a result, nonparametric bootstrap may not well-approximate the distribution of $\\text{Lr}$, while DP provides a better approximation of $\\text{Lr}$, taking advantage of a full sample. \n\\end{remark}\n\n\n\n\n\n\n\n\\section{Theory}\\label{section:theory}\n\nThis section provides the theoretical justification for the proposed methods.\n\n\\subsection{Convergence and consistency of structure learning}\\label{section:consistency}\n\nFirst, we introduce some technical assumptions to derive statistical \nand computational properties of Algorithms \\ref{algorithm:nodewise-regression} and \\ref{algorithm:peeling}. \nLet $(\\bm{\\mathsf Y}, \\bm{\\mathsf X}) \\in \\mathbb R^{n\\times (p+q)}$ be the data matrix. \nDenote by $\\bm\\zeta$ a generic vector and $\\bm\\zeta_A\\in \\mathbb R^{|A|}$ be the projection \nof $\\bm \\zeta$ onto coordinates in $A$. Let \n$\\kappa^\\circ_j = \\|\\bm V^\\circ_{\\cdot j}\\|_0$ and \n$\\kappa^\\circ_{\\max}=\\max_{1\\leq j\\leq p} \\kappa_j^\\circ$.\n\n\\begin{assumption}[Restricted eigenvalues]\\label{assumption:eigen}\n For a constant $c_1>0$, \n\\begin{equation*}\n \\min_{ A : |A|\\leq 2 \\kappa^\\circ_{\\max} }\n\\min_{\\{ \\bm\\zeta : \\|\\bm\\zeta_{A^c}\\|_1\\leq 3\\|\\bm\\zeta_{A}\\|_1 \\}} \n\\frac{\\|\\bm{\\mathsf X} \\bm\\zeta \\|_2^2}{n\\| \\bm\\zeta \\|_2^2} \\geq c_1. \n\\end{equation*}\n\\end{assumption}\n\n\n\\begin{assumption}[Interventions]\\label{assumption:intervention}\n For a constant $c_2>0$, \n\\begin{equation*}\n \\max\\Big(\\max_{1\\leq l\\leq q} n^{-1}(\\bm{\\mathsf X}^\\top\\bm{\\mathsf X})_{ll},\n \\max_{A: |A|\\leq 2 \\kappa^\\circ_{\\max}}\\max_{1\\leq l \\leq |A|} n((\\bm{\\mathsf X}_A^\\top\\bm{\\mathsf X}_A)^{-1})_{ll} \\Big)\n \\leq c_2^2.\n\\end{equation*}\n\\end{assumption}\n\n\\begin{assumption}[Nuisance signals]\\label{assumption:signal}\n \\begin{equation}\\label{equation:signal}\n \\min_{ V^0_{lj} \\neq 0}|V^0_{lj}|\n \\geq 100 \\frac{c_2}{c_1} \\max_{1 \\leq j \\leq p}\\sqrt{\\Omega_{jj}^{-1}\\Big(\\frac{\\log(q)}{n} + \\frac{\\log(n)}{n}\\Big)}.\n \\end{equation}\n\\end{assumption}\n \n\nAssumptions \\ref{assumption:eigen}-\\ref{assumption:intervention}, \nas a replacement of Assumption 1A, are satisfied with a probability tending to one\nfor isotropic subgaussian or bounded $\\bm X$ \n\\citep{rudelson2012reconstruction}. \nAssumption \\ref{assumption:eigen} is a common condition for proving the convergence rate of Lasso regression \n\\citep{bickel2009simultaneous,zhang2014lower}. \nAssumption \\ref{assumption:signal}, as an alternative to Assumption 1B, specifies the minimal signal strength over candidate interventions. \nA signal strength requirement like Assumption \\ref{assumption:signal} is a condition for establishing selection consistency in\nhigh dimensional variable selection \\citep{fan2014strong,loh2017support,zhao2018pathwise}.\nMoreover, Assumption \\ref{assumption:signal} can be further relaxed to a less \nintuitive condition, Assumption \\ref{assumption:relax-signal}; see Appendix \\ref{section:relax-signal} for details.\n\n\n\\begin{theorem} [Finite termination and consistency] \\label{theorem:consistent-structure-learning}\nUnder Assumptions \\ref{assumption:identifiability}-\\ref{assumption:signal} with constants $c_1 < 6c_2$, if the machine precision $\\text{tol}\\ll 1\/n$ is negligible, and tuning parameters of Algorithm \\ref{algorithm:nodewise-regression} satisfy:\n\\begin{enumerate}[(1)]\n \\item $\\gamma_j \\leq c_1\/6$,\n \\item $\\sqrt{32c_2^2\\Omega_{jj}^{-1} (\\log (q) + \\log(n))\/n }\/\\gamma_j \\leq \\tau_j \\leq \n 0.4 \\min_{ V^0_{lj} \\neq 0}|V^0_{lj}|$,\n \\item $\\kappa_j = \\kappa_j^\\circ$,\n\\end{enumerate}\nfor $j=1,\\ldots,p$, \nthen Algorithm \\ref{algorithm:nodewise-regression} yields the global minimizer $\\widehat{\\bm V}_{\\cdot j}$ of \n\\eqref{pseudo-likelihood} in at most $1 + \\lceil \\log (\\kappa_{\\max}^\\circ) \/ \\log 4\\rceil$ DC iterations almost surely, where $\\lceil \\cdot\\rceil$ is the ceiling function. Moreover, \nAlgorithm \\ref{algorithm:peeling} recovers $\\mathcal{A}^\\circ$,\n$\\mathcal C^\\circ \\equiv \\{ l: V^\\circ_{lj}\\neq 0 \\}\\supseteq \\mathcal I^\\circ$, and $\\mathcal D^\\circ$ almost surely as $n\\to\\infty$. \n\\end{theorem}\n\n\n\nBy Theorem \\ref{theorem:consistent-structure-learning} {(proved in Appendix \\ref{proof:consistent-structure-learning})}, \nthe computational complexity of Algorithm \\ref{algorithm:nodewise-regression} is the number of iterations multiplied\nby that of solving a weighted Lasso regression, $O(\\log \\kappa^\\circ_{\\max}) \\times\nO(q^3+n q^2)$ \\citep{efron2004least}. \n{Also note that in Steps 2-6 of Algorithm \\ref{algorithm:peeling}, no heavy computation is involved. Thus, the overall time complexity of Algorithm \\ref{algorithm:peeling} is bounded by $O(p\\times \\log \\kappa^\\circ_{\\max} \\times (q^3+n q^2) )$ for computing $\\widehat{\\bm V}$ matrix.}\nFinally, the peeling algorithm does not apply to observational data ($\\bm W = \\bm 0$). In a sense, interventions play an essential role.\n\nAlthough Theorem \\ref{theorem:consistent-structure-learning} establishes\nthe consistent reconstruction of the peeling algorithm \nfor the super-graph $S$, it does not provide any uncertainty \nmeasure of the presence of some directed edges of the true DAG. \nIn what follows, we will develop a theory for hypothesis tests to make valid inference concerning directed edges of interest. \n\n\n\n\n\\subsection{Inferential theory}\\label{section:inference-theory}\n\n\\begin{assumption}[Hypothesis-specific dimension restriction]\\label{assumption:dimension}\nFor a constant $\\rho$, \n\\begin{equation*}\n \\max_{j : \\textnormal{\\textsc{d}}_{G^\\circ}(j)\\neq \\emptyset} \\frac{|\\textnormal{\\textsc{an}}_{G^\\circ}(j)| + |\\textnormal{\\textsc{d}}_{G^\\circ}(j)| +{\\kappa_j^\\circ}}{n} \\leq \\rho < 1, \\quad \\text{ as } n \\rightarrow \\infty.\n\\end{equation*}\n\\end{assumption}\n\n\n Assumption \\ref{assumption:dimension} is a hypothesis-specific condition restricting the underlying dimension of the testing problem. \n Usually, $|\\textnormal{\\textsc{an}}_{G^\\circ}(j)| \\asymp \\kappa_j^\\circ \\ll p$; $j=1,\\ldots,p$, \nwhich dramatically relaxes the condition \n$n\\gg {|\\mathcal D|^{1\/2} p\\log(p)}$ for \nthe constrained likelihood ratio test \n\\citep{li2019likelihood}. \n\n\n\n\\begin{theorem}[Empirical p-values]\n \\label{theorem:dp-null-distribution}\nSuppose Assumptions \\ref{assumption:identifiability}-\\ref{assumption:dimension} are met and $H_0$ is regular. \nIf the tuning parameters in Algorithm \\ref{algorithm:nodewise-regression}\nsatisfy (1)-(3) in Theorem \\ref{theorem:consistent-structure-learning}, then, \n \\begin{enumerate}[(A)]\n \\item (Test of directed edges \\eqref{equation:link-test}) \n \\begin{equation*}\n \\begin{split} \n \\lim_{\n \\substack{n\\to\\infty\\\\\n \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:link-test}}\n } \n \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} < \\alpha) = \\alpha, \n &\\quad \\text{ if } H_0 \\text{ is nondegenerate};\\\\\n \\lim_{\n \\substack{n\\to\\infty\\\\\n \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:link-test}}\n } \n \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} = 1) = 1, &\\quad \\text{ if } H_0 \\text{ is degenerate}.\n \\end{split}\n \\end{equation*}\n \\item (Test of directed pathways \\eqref{equation:pathway-test})\n \\begin{equation*}\n \\begin{split}\n \\limsup_{\n \\substack{n\\to\\infty\\\\\n \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:pathway-test}}\n } \n \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} < \\alpha) = \\alpha, &\\quad \\text{ if } H_0 \\text{ is nondegenerate with } |\\mathcal D|=|\\mathcal H|;\\\\\n \\limsup_{\n \\substack{n\\to\\infty\\\\\n \\bm \\theta^\\circ \\textnormal{ satisfies H}_0 \\textnormal{ in } \\eqref{equation:pathway-test}}\n } \n \\lim_{M \\rightarrow \\infty} \\operatorname{\\mathbb P}_{\\bm\\theta^\\circ}(\\textnormal{Pval} = 1) = 1, &\\quad \\text{ if } |\\mathcal D|<|\\mathcal H|.\n \\end{split}\n \\end{equation*}\n \\end{enumerate}\n\\end{theorem} \n\nBy Theorem \\ref{theorem:dp-null-distribution}, the DP likelihood ratio test yields a valid p-value for \\eqref{equation:link-test} and \\eqref{equation:pathway-test}.\nNote that $|\\mathcal D|$ is permitted to depend on $n$. \nMoreover, Proposition \\ref{proposition:asymptotic} summarizes the asymptotics for \ndirected edge test \\eqref{equation:link-test}. \n\n\\begin{proposition} [Asymptotic distribution of $\\textnormal{Lr}$ of \\eqref{equation:link-test}] \\label{proposition:asymptotic}\n Assume that the assumptions of Theorem \\ref{theorem:dp-null-distribution} are met. Under a regular $H_0$, as $n\\to\\infty$, \n \\begin{equation*}\n \\begin{split}\n 2 \\textnormal{Lr} \\overset{d}{\\longrightarrow} \\chi^2_{|\\mathcal D|}, & \\text{ if } H_0 \\text{ is nondegenerate with } |\\mathcal D| > 0 \\text{ being fixed}; \\\\\n \\frac{2 \\textnormal{Lr} - |\\mathcal D|}{\\sqrt{2|\\mathcal D|}} \\overset{d}{\\longrightarrow} N(0,1), & \\text{ if } H_0 \\text{ is nondegenerate with } |\\mathcal D| \\rightarrow \\infty, \n \\ \\frac{|\\mathcal D|\\log |\\mathcal D|}{n} \\rightarrow 0.\n \\end{split}\n\\end{equation*}\n\\end{proposition}\n\n{The proofs of Theorem \\ref{theorem:dp-null-distribution} and Proposition \\ref{proposition:asymptotic} are given in Appendix \\ref{proof:dp-null-distribution} and \\ref{proof:asymptotic}.}\n\n\\begin{remark}[Local structure for hypothesis testing]\n Theorem \\ref{theorem:dp-null-distribution} or Proposition \\ref{proposition:asymptotic} only requires correct identification of the local structures of the DAG such as, $\\textnormal{\\textsc{an}}_S(j) = \\textnormal{\\textsc{an}}_G(j)$, $\\textnormal{\\textsc{d}}_S(j) = \\textnormal{\\textsc{d}}_G(j)$, and $\\textnormal{\\textsc{in}}_S(j)$, for variables $Y_j$ with $\\textnormal{\\textsc{d}}_S(j)\\neq\\emptyset$, as opposed to those of entire $S$.\n\\end{remark}\n\n\nNext, we analyze the local limit power of the proposed tests for \\eqref{equation:link-test} and \\eqref{equation:pathway-test}. \nAssume $\\bm \\theta^\\circ = (\\bm U^\\circ,\\bm W^\\circ)$ satisfies $H_0$. \nLet $\\bm \\Delta\\in\\mathbb R^{p\\times p}$ satisfy\n$\\bm \\Delta_{\\mathcal D^c}=\\bm 0$ so that $\\bm U^\\circ+\\bm \\Delta$ represents a DAG. \nFor nondegenerate and regular $H_0$, consider an alternative $H_a$: $\\bm U_{\\mathcal H} = \\bm U_{\\mathcal H}^\\circ + \\bm \\Delta_{\\mathcal H}$,\nand define the power function as \n\\begin{equation}\n \\beta(\\bm\\theta^\\circ,\\bm \\Delta) = \\operatorname{\\mathbb P}_{H_a}( \\text{Pval} < \\alpha ).\n\\end{equation}\n\n\n\n\\begin{proposition}[Local power of edge test]\n\\label{proposition:power-edge}\nAssume that $H_0$ is nondegenerate and regular. \nLet $\\| \\bm \\Delta\\|_{F} = \n\\|\\bm \\Delta_{\\mathcal H}\\|_{F} = n^{-1\/2} \\delta$ when $|\\mathcal D| > 0$ is fixed \nand $\\|\\bm \\Delta\\|_{F} = |\\mathcal D|^{1\/4} n^{-1\/2}h$ when $|\\mathcal D|\\to\\infty$, \nwhere $\\delta>0$ and $\\|\\cdot\\|_{F}$ is the matrix Frobenius norm. \nIf the assumptions of Theorem \\ref{theorem:dp-null-distribution} are met, then under $H_a$, as $n,M\\to\\infty$,\n\\begin{equation*}\n \\beta(\\bm\\theta^\\circ,\\bm \\Delta) \\geq\n \\begin{cases}\n \\operatorname{\\mathbb P}\\Big(\\|\\bm Z + c_l\\sqrt{n}\\bm \\Delta \\|_2^2 > \\chi^2_{|\\mathcal D|,1-\\alpha} \\Big) & \\text{if } |\\mathcal D|>0 \\text{ is fixed};\\\\\n \\operatorname{\\mathbb P}\\Big(Z > z_{1-\\alpha} - {c_l \\|\\bm \\Delta\\|^2_2}\/{\\sqrt{2|\\mathcal D|}} \\Big) & \\text{if } |\\mathcal D|\\to\\infty, \\frac{|\\mathcal D|\\log|\\mathcal D|}{n}\\to 0,\n \\end{cases}\n\\end{equation*}\nwhere $\\bm Z\\sim N(\\bm 0,\\bm I_{|\\mathcal D|})$, $Z\\sim N(0,1)$, and $\\chi^2_{|\\mathcal D|,1-\\alpha}$ and $z_{1-\\alpha}$ are the $(1-\\alpha)$th quantile of distributions $\\chi^2_{|\\mathcal D|}$ and $N(0,1)$, respectively.\nIn particular, $\\lim_{\\delta\\to\\infty} \\lim_{n\\to\\infty} \\beta(\\bm\\theta^\\circ,\\bm \\Delta) = 1$.\n\\end{proposition}\n\n\\begin{proposition} [Local power of pathway test]\n \\label{proposition:power-path}\nAssume that $H_0$ is nondegenerate and regular with $|\\mathcal D| = |\\mathcal H|$. \nLet $\\min_{(k,j)\\in \\mathcal H}|U^\\circ_{kj} + \\Delta_{kj}| = n^{-1\/2}\\delta$ when $|\\mathcal H|>0$ is fixed\nand $\\min_{(k,j)\\in \\mathcal H}|U^\\circ_{kj} + \\Delta_{kj}| = n^{-1\/2}\\delta\\sqrt{\\log|\\mathcal H|}$ when $|\\mathcal H|\\to\\infty$.\nIf the assumptions of Theorem \\ref{theorem:dp-null-distribution} are met, then under $H_a$, as $n,M\\to\\infty$,\n \\begin{equation*}\n \\beta(\\bm\\theta^\\circ,\\bm \\Delta) \\geq 1 - \\frac{|\\mathcal H|}{\\sqrt{2\\pi}} \n \\exp\\Big( -\\Big(\\delta\\sqrt{\\log|\\mathcal H|}\/\\max_{1\\leq j\\leq p}\\Omega_{jj} - \\sqrt{\\chi^2_{1,1-\\alpha}}\\Big)^2\/2 \\Big),\n \\end{equation*}\n where $Z \\sim N\\Big(\\delta^2\/\\max_{1\\leq j\\leq p}\\Omega_{jj},1\\Big)$ and \n $\\chi^2_{1,1-\\alpha}$ is the $(1-\\alpha)$th quantile of distribution $\\chi^2_{1}$.\n Then, $\\lim_{\\delta\\to\\infty} \\lim_{n\\to\\infty} \\beta(\\bm\\theta^\\circ,\\bm \\Delta) = 1$.\n\\end{proposition}\n\n{The proofs of Propositions \\ref{proposition:power-edge} and \\ref{proposition:power-path} are deferred to Appendix \\ref{proof:power-edge} and \\ref{proof:power-path}.}\n\n\\section{Simulations}\\label{section:simulation}\n\nThis section investigates the operating characteristics of the proposed tests and the peeling algorithm via simulations. In simulations, we consider two setups for generating $\\bm U\\in\\mathbb{R}^{p\\times p}$, representing random and hub DAGs, respectively.\n\\begin{itemize}\n \\item \\textbf{Random graph.} The upper off-diagonal entries $U_{kj}$; $k0$ and\n$n>>0$. If $\\mathcal{L}$ is a $\\sigma$-ample invertible sheaf on $V$, then\n\\begin{equation}\\label{eq2.2}\nMod~(B(V, \\mathcal{L}, \\sigma)) \/ ~Tors ~\\cong ~Coh~(V),\n\\end{equation}\nwhere $Mod$ is the category of graded left modules over the ring $B(V, \\mathcal{L}, \\sigma)$,\n$Tors$ is the full subcategory of $Mod$ of the torsion modules and $Coh$ is the category of \nquasi-coherent sheaves on a scheme $V$. In other words, the $B(V, \\mathcal{L}, \\sigma)$ is \na coordinate ring of the variety $V$.\n\\begin{example}\\label{ex2.1}\n([Stafford \\& van den Bergh 2001] \\cite[p.173]{StaVdb1})\nDenote by $P^1(k)$ a projective line over the field $k$.\nConsider an automorphism $\\sigma$ of the $P^1(k)$\ngiven by the formula $\\sigma(u)=qu$, where $u\\in P^1(k)$\nand $q\\in k^{\\times}$. Then $B(P^1(k), \\mathcal{L}, \\sigma)\\cong U_q$,\nwhere $U_q$ is the $k$-algebra of polynomials in \nvariables $x_1$ and $x_2$ satisfying a commutation relation:\n\\begin{equation}\\label{eq2.3}\nx_2x_1=qx_1x_2.\n\\end{equation}\n\\end{example}\n\\begin{example}\n([Stafford \\& van den Bergh 2001] \\cite[p.197]{StaVdb1})\nDenote by \n$\\mathcal{E}(k)=\\{(u,v,w,z)\\in P^3(k) ~|~u^2+v^2+w^2+z^2 =\n{1-\\alpha\\over 1+\\beta}v^2+{1+\\alpha\\over 1-\\gamma}w^2+z^2 = 0\\}$ \nan elliptic curve over the field $k$, where $\\alpha,\\beta,\\gamma\\in k$\nare constants, such that $\\beta\\ne -1$ and $\\gamma\\ne 1$.\nLet $\\sigma$ be a shift automorphism of the $\\mathcal{E}(k)$.\nThen $B(\\mathcal{E}(k), \\mathcal{L}, \\sigma)\\cong S(\\alpha,\\beta,\\gamma)$,\nwhere $S(\\alpha,\\beta,\\gamma)$ is the Sklyanin algebra on four generators\n$x_i$ satisfying the commutation relations:\n\n\\begin{equation}\\label{eq2.4}\n\\left\\{\n\\begin{array}{ccc}\nx_1x_2-x_2x_1 &=& \\alpha(x_3x_4+x_4x_3),\\\\\nx_1x_2+x_2x_1 &=& x_3x_4-x_4x_3,\\\\\nx_1x_3-x_3x_1 &=& \\beta(x_4x_2+x_2x_4),\\\\\nx_1x_3+x_3x_1 &=& x_4x_2-x_2x_4,\\\\\nx_1x_4-x_4x_1 &=& \\gamma(x_2x_3+x_3x_2),\\\\ \nx_1x_4+x_4x_1 &=& x_2x_3-x_3x_2,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\alpha+\\beta+\\gamma+\\alpha\\beta\\gamma=0$. \n\\end{example}\n\\begin{example}\\label{ex2.3}\n(\\cite[Lemma 3.1]{Nik2})\nLet $\\mathscr{R}$ be an arithmetic Riemann surface, i.e. given by the \nAF-algebra of stationary type \\cite[Section 5.2]{N}. \n(Such Riemann surfaces can be identified with the complex \nalgebraic curves defined over a number field.). Then \n\\begin{equation}\nB(\\mathscr{R}, \\mathcal{L}, \\sigma)\\cong R[\\pi_1(S^3 \\backslash \\mathscr{L})],\n\\end{equation}\nwhere $\\mathscr{L}$ is a link embedded in the three-sphere $S^3$ \nand $R[\\pi_1(S^3 \\backslash \\mathscr{L})]$\nis the group ring of the fundamental group $\\pi_1(S^3 \\backslash \\mathscr{L})$. \n\\end{example}\n\n\n\\subsection{Arithmetic groups}\nLet $G$ be a linear algebraic group defined over the field $\\mathbf{Q}$.\nDenote by $G_{\\mathbf{Z}}$ the group of integer points of $G$. A subgroup \n$\\Gamma\\subset G$ is called {\\it arithmetic} if $\\Gamma$ is commensurable with \nthe $G_{\\mathbf{Z}}$, i.e. $\\Gamma\\cap G_{\\mathbf{Z}}$ has a finite index both\nin $\\Gamma$ and $G_{\\mathbf{Z}}$. Informally, the arithmetic group is a discrete \nsubgroup of the group $GL_n(\\mathbf{C})$ defined by some arithmetic properties.\nFor instance, $\\mathbf{Z}\\subset\\mathbf{R}$, $GL_n(\\mathbf{Z})\\subset GL_n(\\mathbf{R})$ and\n$SL_n(\\mathbf{Z})\\subset SL_n(\\mathbf{R})$ are examples of the arithmetic groups. \n\n\nDenote by $\\mathcal{O}$ the ring of algebraic integers of all finite \nextensions of the number field $\\mathbf{Q}$. \nLet $\\mathbb{H}^3$ be the hyperbolic 3-dimensional space.\nThe following remarkable result establishes \na deep link between arithmetic groups and topology.\n\\begin{theorem}\\label{thm2.4}\n{\\bf ([Maclachlan \\& Reid 2003] \\cite[p. 169]{MR})}\nLet $M=\\mathbb{H}^3\/\\Gamma$ be a finite volume hyperbolic 3-manifold.\nThen $\\Gamma$ is conjugate to a subgroup of the group $PSL_2(\\mathcal{O})$. \n\\end{theorem}\n\\begin{example}\\label{ex2.5}\nLet $\\mathscr{L}$ be a hyperbolic link, i.e. $S^3\\backslash\\mathscr{L}\\cong \\mathbb{H}^3\/\\Gamma$\nfor an arithmetic group $\\Gamma$. Then \n\\begin{equation}\\label{eq2.5} \n\\pi_1(S^3 \\backslash\\mathscr{L})\\cong\\Gamma.\n\\end{equation}\n\\end{example}\n\n\n\n\\section{Proof of theorem \\ref{thm1.3}}\n(i) Let us show that the $\\mathbf{C}P^1$ is a karma of $\\mathbf{Z}$.\nIndeed, in this case $R\\cong\\mathbf{Z}$ and $\\mathscr{A}_{\\mathbf{Z}}$ is the closure of a self-adjoint representation of\nthe ring $M_2(\\mathbf{Z})$. Consider the group $PSL_2(\\mathbf{Z})=SL_2(\\mathbf{Z})\/\\pm I$, where $SL_2(\\mathbf{Z})$ is the group \nof invertible elements of $M_2(\\mathbf{Z})$. \nRecall that the group $PSL_2(\\mathbf{Z})$ is generated by the matrices: \n\\begin{equation}\\label{eq3.1}\nu=\\left(\n\\begin{matrix}\n0 & 1\\cr -1 & 0\n\\end{matrix}\n\\right)\n\\quad\n\\hbox{and}\n \\quad v=\\left(\n\\begin{matrix}\n0 & -1\\cr 1 & -1\n\\end{matrix}\n\\right)\n\\end{equation}\nwhich satisfy the relations modulo $\\pm I$:\n\\begin{equation}\\label{eq3.2}\nu^2= v^3=1.\n\\end{equation}\n\n\n\\bigskip\nOn the other hand, consider Example \\ref{ex2.1} with $k\\cong\\mathbf{Q}$ and assume \nthat $q=-1$ in relation (\\ref{eq2.3}). In other words, one gets a relation:\n\\begin{equation}\\label{eq3.3}\nx_2x_1=-x_1x_2.\n\\end{equation}\n\nConsider a substitution:\n\\begin{equation}\\label{eq3.4}\n\\left\\{\n\\begin{array}{ccl}\nu&=& x_2x_1x_2^{-1}x_1^{-1}\\\\\nv &=& x_2. \n\\end{array}\n\\right.\n\\end{equation}\n\nThe reader can verify, that substitution (\\ref{eq3.4}) and relation (\\ref{eq3.3}) \nreduces relations (\\ref{eq3.2}) to the form: \n\\begin{equation}\\label{eq3.5}\nx_2^3=1.\n\\end{equation}\n\nLet $\\mathscr{I}$ be a two-sided ideal in the algebra $B(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma)$ of \nExample \\ref{ex2.1} generated by relation (\\ref{eq3.5}). In view of (\\ref{eq3.2})-(\\ref{eq3.5}), one gets a\nring isomorphism: \n\\begin{equation}\\label{eq3.6}\nB(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma)\/\\mathscr{I}\\cong M_2(\\mathbf{Z}). \n\\end{equation}\n\nLet $\\rho$ be a self-adjoint representation of the ring $B(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma)$\nby the linear operators on a Hilbert space $\\mathscr{H}$. Notice that such a representation \nexists, because relation (\\ref{eq3.3}) is invariant under the involution $x_1^*=x_2$ and \n$x_2^*=x_1$. Since $\\rho(B(P^1(\\mathbf{Q}), \\mathcal{L}, \\sigma))=\\mathscr{A}_{P^1(\\mathbf{Q})}$\nand $\\rho(M_2(\\mathbf{Z}))=\\mathscr{A}_{\\mathbf{Z}}$, it follows from (\\ref{eq3.6}) that there exists a \n$C^*$-algebra homomorphism\n\\begin{equation}\\label{eq3.7}\nh: \\mathscr{A}_{P^1(\\mathbf{Q})}\\to\\mathscr{A}_{\\mathbf{Z}},\n\\end{equation}\nwhere $Ker ~h=\\rho(\\mathscr{I})$. \nThe homomorphism $h$ extends to a homomorphism between the \n products\n\\begin{equation}\\label{eq3.8}\nh: \\mathscr{A}_{P^1(\\mathbf{Q})}\\otimes\\mathscr{K}\\to\\mathscr{A}_{\\mathbf{Z}}\\otimes\\mathscr{K}, \n\\end{equation}\nwhere $\\mathscr{K}$ is the $C^*$-algebra of compact operators. \nBut $\\mathscr{A}_{P^1(\\mathbf{Q})}\\otimes\\mathscr{K}\\cong \\mathscr{A}_{\\mathbf{C}P^1}$\nand, therefore, one gets a $C^*$-algebra homomorphism\n\\begin{equation}\\label{eq3.9}\nh: \\mathscr{A}_{\\mathbf{C}P^1} \\to\\mathscr{A}_{\\mathbf{Z}}\\otimes\\mathscr{K}. \n\\end{equation}\n\nIn other words, the Riemann sphere $\\mathbf{C}P^1$ is a karma of the ring $\\mathbf{Z}$. \n\n\n\n\n\n\n\\bigskip\n(ii) Let us show that if $K$ is a number field, then there exists a Riemann surface $\\mathscr{R}$, \nsuch that $\\mathscr{R}$ is a karma of the ring $O_K$. \nIndeed, we can always assume that $K$ has at least one complex embedding and fix one of such embeddings \n$K\\not\\subset\\mathbf{R}$. \n (For otherwise, we replace $K$ by a CM-field of $K$, i.e. a totally imaginary quadratic extension\nof the totally real field $K$. This case corresponds to the double covering $\\mathscr{R}'$of the Riemann surface $\\mathscr{R}$.) \nFor simplicity, let $R\\cong O_K$ and $\\Gamma\\cong PSL_2(O_K)$. (The case of a non-maximal order $\\Lambda\\subseteq O_K$ is treated likewise and corresponds to the covering of \nthe Riemann surface $\\mathscr{R}$.) In view of (\\ref{eq2.5}), there exists a hyperbolic link $\\mathscr{L}$, \nsuch that: \n\\begin{equation}\\label{eq3.10} \nPSL_2(O_K)\\cong\\pi_1(S^3 \\backslash\\mathscr{L}).\n\\end{equation}\n\n\\bigskip\nOn the other hand, it is known that\n\\begin{equation}\\label{eq3.11} \nR[\\pi_1(S^3 \\backslash\\mathscr{L})]\\cong B(\\mathscr{R}, \\mathcal{L}, \\sigma), \n\\end{equation}\nwhere $R[\\pi_1(S^3 \\backslash\\mathscr{L})]$ is the group ring of $\\pi_1(S^3 \\backslash\\mathscr{L})$ and \n $\\mathscr{R}$ is a Riemann surface, see example \\ref{ex2.3}. In particular, it follows from (\\ref{eq3.10}) \n that \n\\begin{equation}\\label{eq3.12} \nB(\\mathscr{R}, \\mathcal{L}, \\sigma)\\cong R[PSL_2(O_K)]. \n\\end{equation}\n\n\nLet $\\rho$ be a self-adjoint representation of the ring $B(\\mathscr{R}, \\mathcal{L}, \\sigma)$\nby the linear operators on a Hilbert space $\\mathscr{H}$. \nThe norm closure of $\\rho(B(\\mathscr{R}, \\mathcal{L}, \\sigma))$\nis the Serre $C^*$-algebra $\\mathscr{A}_{\\mathscr{R}}$. \n\n\nOn the other hand, it follows from (\\ref{eq3.12}) that taking the norm closure of \n$\\rho(R[PSL_2(O_K)])$, one gets a $C^*$-algebra $\\mathscr{A}_{O_K}$, \nsuch that \n\\begin{equation}\\label{eq3.13} \n\\mathscr{A}_{O_K}\\otimes\\mathscr{K}\\cong \\mathscr{A}_{\\mathscr{R}}. \n\\end{equation}\n\nIn other words, there exists an isomorphism:\n\\begin{equation}\\label{eq3.14} \nh: \\mathscr{A}_{\\mathscr{R}}\\to \\mathscr{A}_{O_K}\\otimes\\mathscr{K}. \n\\end{equation}\n\nIt follows from (\\ref{eq3.14}) that the Riemann surface $\\mathscr{R}$ is a karma of \nthe ring $O_K$. \n\n\n\\bigskip\n(iii) Finally, let us show that \n the inclusion $\\mathbf{Z}\\subset O_K$ defines a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three points $\\{0,1, \\infty\\}$. \n\n\nIn the lemma below we shall prove a stronger result. \nNamely, let $\\mathfrak{K}$ be a category of the Galois extensions of the field $\\mathbf{Q}$,\nsuch that the morphisms in $\\mathfrak{K}$ are inclusions $K\\subseteq K'$, where $K,K'\\in\\mathfrak{K}$. \n Likewise, let $\\mathfrak{R}$ be a category of the Riemann surfaces,\nsuch that the morphisms in $\\mathfrak{R}$ are holomorphic maps $\\mathscr{R}\\to \\mathscr{R}'$, \nwhere $\\mathscr{R}, \\mathscr{R}'\\in\\mathfrak{R}$. \nLet $F: \\mathfrak{K}\\to\\mathfrak{R}$ be a map acting by the formula $O_K\\mapsto\\mathscr{R}$, where\n$\\mathscr{R}$ is the Riemann surface defined by the isomorphism (\\ref{eq3.12}). \n\\begin{remark}\\label{rmk3.1}\nThe category $\\mathfrak{R}$ consists of the Riemann surfaces, which are algebraic curves defined over a number field.\nIn particular, the morphisms in $\\mathfrak{R}$ can be defined over the number field. Both facts follow from \nthe property of the AF-algebra $\\mathscr{A}_{\\mathscr{R}}$ being of a stationary type \\cite[Section 5.2]{N}. \nWe refer the reader to Example \\ref{ex2.3} and \\cite[Lemma 3.1]{Nik2}. \n\\end{remark}\n\\begin{lemma}\\label{lm3.1}\nThe map $F: \\mathfrak{K}\\to\\mathfrak{R}$ is a covariant functor, i.e. $F$ transforms inclusions \nin the category $\\mathfrak{K}$ to holomorphic maps in the category $\\mathfrak{R}$. \n\\end{lemma}\n\\begin{proof}\nLet $K\\in\\mathfrak{K}$ be a number field and let $\\mathscr{R}=F(K)$ be the corresponding\nRiemann surface $\\mathscr{R}\\in\\mathfrak{R}$. Let $K\\subseteq K'$ be an inclusion, where \n$K'\\in \\mathfrak{K}$.\n\nUsing isomorphism (\\ref{eq3.13}), one gets an inclusion of the corresponding Serre $C^*$-algebras: \n\\begin{equation}\n\\mathscr{A}_{\\mathscr{R}}\\subseteq \\mathscr{A}_{\\mathscr{R}'}. \n\\end{equation}\n\nOn the other hand, it is known the algebra $\\mathscr{A}_{\\mathscr{R}}$\nis a coordinate ring of the Riemann surface $\\mathscr{R}$ \\cite[Theorem 5.2.1]{N}. \n In particular, if $h: \\mathscr{A}_{\\mathscr{R}'}\\to \\mathscr{A}_{\\mathscr{R}}$ is a homomorphism,\n one gets a holomorphic map $w: \\mathscr{R}'\\to\\mathscr{R}$ defined by a commutative diagram in Figure 1. \n\\begin{figure}[h]\n\\begin{picture}(300,110)(-70,-5)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(130,70){\\vector(0,-1){35}}\n\\put(45,23){\\vector(1,0){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(13,20){$ \\mathscr{A}_{\\mathscr{R}'}$}\n\\put(75,30){$h$}\n\\put(75,90){$w$}\n\\put(123,20){$ \\mathscr{A}_{\\mathscr{R}}$}\n\\put(17,80){$\\mathscr{R}'$}\n\\put(122,80){$\\mathscr{R}$}\n\\end{picture}\n\\caption{Holomorphic map $w$.}\n\\end{figure}\n\n\nThus $F$ is a functor, which maps the inclusion $K\\subseteq K'$\ninto a holomorphic map $w: \\mathscr{R}'\\to\\mathscr{R}$. The reader \ncan verify that $F$ is a covariant functor. Lemma \\ref{lm3.1} is proved. \n\\end{proof}\n\n\\begin{lemma}\\label{lm3.3}\n The inclusion $\\mathbf{Z}\\subset O_K$ defines a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three points $\\{0,1, \\infty\\}$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathscr{U}$ be the Riemann sphere $\\mathbf{C}P^1$ without three points,\nwhich we always assume to be $\\{0,1,\\infty\\}$ after a proper M\\\"obius transformation. \nIt is easy to see, that the fundamental group $\\pi_1(\\mathscr{U})\\cong \\mathfrak{F}_2$, where \n $\\mathfrak{F}_2$ is a free group on two generators $u$ and $v$. \n \n \n Since the the Riemann surface $\\mathscr{U}$ corresponds to \n an unlink $\\mathscr{L}\\cong S^1\\cup S^1$, one gets an isomorphism:\n\n \\begin{equation}\\label{eq3.16}\n B(P^1(\\mathscr{U}, \\mathcal{L}, \\sigma))\\cong R[\\mathfrak{F}_2].\n \\end{equation}\n\n\n\n\n\\smallskip\nConsider a two-sided ideal $\\mathscr{I}\\subset B(P^1(\\mathscr{U}, \\mathcal{L}, \\sigma))$\ngenerated by relations (\\ref{eq3.2}). In view of (\\ref{eq3.16}), we have:\n\n \\begin{equation}\\label{eq3.17}\n B(P^1(\\mathscr{U}, \\mathcal{L}, \\sigma))\/\\mathscr{I}\\cong R[PSL_2(\\mathbf{Z})].\n \\end{equation}\n\n\n\\smallskip\nIn other words, one gets a homomorphism between the $C^*$-algebras:\n \\begin{equation}\\label{eq3.18}\n \\mathscr{A}_{\\mathscr{U}}\\to \\mathscr{A}_{\\mathbf{C}P^1}. \n \\end{equation}\n\n\nUsing the commutative diagram in Figure 1, we get a holomorphic map between \nthe corresponding Riemann surfaces:\n \\begin{equation}\\label{eq3.19}\n \\mathscr{U}\\to \\mathbf{C}P^1. \n \\end{equation}\n\n\n\n \n \n\n\n \n \\bigskip \n Let now $\\mathbf{Z}\\subset O_K$ be an inclusion, where $K$ is a number field. \n By item (ii) of theorem \\ref{thm1.3} there exists a Riemann surface $\\mathscr{R}\\in\\mathfrak{R}$ \n corresponding to $O_K$. By lemma \\ref{lm3.1}, there exists a holomorphic map:\n \\begin{equation}\\label{eq3.20}\n \\mathscr{R}\\to \\mathbf{C}P^1. \n \\end{equation}\n\n \nUsing (\\ref{eq3.19}) and (\\ref{eq3.20}), one gets a commutative digram in Figure 2. \n\\begin{figure}[h]\n\\begin{picture}(300,90)(-90,10)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(45,30){\\vector(3,2){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(13,20){$\\mathscr{U}$}\n\\put(17,80){$\\mathscr{R}$}\n\\put(122,80){$\\mathbf{C}P^1$}\n\\end{picture}\n\\caption{The map $\\mathscr{R}\\to \\mathscr{U}$.}\n\\end{figure}\n\n\n \n \\medskip\nWe use the diagram in Figure 2 to define a holomorphic map:\n\n \\begin{equation}\\label{eq4.4}\n \\mathscr{R}\\to \\mathscr{U}.\n \\end{equation}\n\n \n \\smallskip\n Since $\\mathscr{U}=\\mathbf{C}P^1\\backslash\\{0,1,\\infty\\}$, one gets the conclusion of \n lemma \\ref{lm3.3}. \n \\end{proof}\n\n\n\\bigskip\nItem (iii) of theorem \\ref{thm1.3} follows from lemmas \\ref{lm3.1} and \\ref{lm3.3}. \n\n\n\n\\bigskip\nTheorem \\ref{thm1.3} is proved. \n\n \n\n\\section{Belyi's Theorem}\nBelyi's Theorem says that the algebraic curve $\\mathscr{R}$ can be defined over \na number field if and only if there exist a covering $\\mathscr{R}\\to \\mathbf{C}P^1$\nramified over three points of the Riemann sphere \n$\\mathbf{C}P^1$. This remarkable result was proved by [Belyi 1979] \\cite[Theorem 4]{Bel1}. \nIn this section we show that Belyi's Theorem follows from theorem \\ref{thm1.3} and remark \\ref{rmk3.1}. \n\\begin{theorem}\\label{thm4.1}\n\\textbf{(Belyi's Theorem)}\nA complete non-singular algebraic curve over $\\mathbf{C}$\n can be defined over an algebraic number field if and only if such a curve is a covering\nof the Riemann sphere $\\mathbf{C}P^1$ ramified over three points. \n\\end{theorem}\n\\begin{proof}\nWe identify the Riemann surface $\\mathscr{R}\\in\\mathfrak{R}$ with a complete non-singular algebraic \ncurve over the field of characteristic zero (Chow's Theorem). \n\n\\smallskip\nIn view of the remark \\ref{rmk3.1}, each $\\mathscr{R}\\in\\mathfrak{R}$ is the algebraic curve\n defined over a finite extension of the field $\\mathbf{Q}$. \n On the other hand, item (iii) of theorem \\ref{thm1.3} says that each Riemann surface \n$\\mathscr{R}\\in\\mathfrak{R}$ is a covering of the $\\mathbf{C}P^1$ ramified over the points\n$\\{0,1,\\infty\\}$. The ``only if'' part of Belyi's Theorem follows. \n \n\\smallskip \nLet $\\mathscr{R}$ be a covering of the $\\mathbf{C}P^1$ ramified over the points\n$\\{0,1,\\infty\\}$. Using lemma \\ref{lm3.1}, one can construct a ring $O_K$ corresponding\nto the Riemann surface $\\mathscr{R}$. By item (ii) of theorem \\ref{thm1.3} and remark \\ref{rmk3.1} we have\n$\\mathscr{R}\\in\\mathfrak{R}$. In other words, $\\mathscr{R}$ is an algebraic curve defined over an algebraic number field. \nThe ``if'' part of of Belyi's Theorem is proved.\n \\end{proof}\n\\begin{remark}\nIt is an interesting problem to calculate the ramification data and equations of the Belyi curves $\\mathscr{R}$\nin terms of the orders $\\Lambda\\subseteq O_K$ and number fields $K$ obtained in theorem \\ref{thm1.3}. \nHowever this problem is out of the scope of present paper. \n\\end{remark}\n\n\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}