diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzqhoz" "b/data_all_eng_slimpj/shuffled/split2/finalzzqhoz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzqhoz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{intro}\nQuantum steering was first proposed by Schr\\\"odinger in 1936 \\cite{Schrodinge1936} in response to the EPR paradox \\cite{einstein1935can}. However, it did not attract much attentions until 2007, when Wiseman \\textit{et al.} reinterpreted quantum steering strictly from the operational view and even proposed some experimental criteria \\cite{wiseman2007steering}. Ever since, the research of quantum steering has made great progress both in theory \\cite{piani2015necessary,sun2017exploration} and experiment \\cite{kocsis2015experimental,cavalcanti2016quantum,deng2017demonstration,zhao2020experimental,wollmann2020experimental}. In Wiseman's definition, quantum steering that logically intermediates between quantum entanglement and Bell nonlocality, describes the ability of one party, Alice, to nonlocally control the state of another party, Bob, even when Bob does not trust Alice's measurement apparatus, exhibiting unique asymmetric behavior \\cite{gallego2015resource,he2013genuine,xiao2017demonstration,uola2020quantum}. As an essential type of quantum correlations, quantum steering has great applications in quantum key distribution \\cite{gehring2015implementation,walk2016experimental}, subchannel discrimination \\cite{sun2018demonstration}, asymmetric quantum network \\cite{cavalcanti2015detection}, randomness generation \\cite{skrzypczyk2018maximal,guo2019experimental} and randomness certification \\cite{curchod2017unbounded}. \nIn the standard EPR steering tasks, $N$ entangled particles are separately distributed to $N$ different observers and each observer performs some projective (sharp) measurements to demonstrate her or his steerability. Since each observer is spatially separated, the non-signaling condition is strictly satisfied between different observers, i.e., the marginal probability distribution of each observer does not depend on the measurements of any other observers \\cite{masanes2006general}. Due to the monogamy constraints, the number of observers who share quantum correlation via sharp measurement is limited \\cite{coffman2000distributed, toner2006monogamy,reid2013monogamy,mal2017necessary}. Recently, a surprising result was reported by Silva \\textit{et al.} that the number of observers sharing non-locality can be increased if the sequential weak (unsharp) measurement was employed, where the non-signaling condition is dropped \\cite{silva2015multiple}. Their result later is confirmed by theoretical \\cite{mal2016sharing,das2019facets,brown2020arbitrarily} as well as experimental works \\cite{schiavon2017three,hu2018observation} and the sequential unsharp measurement strategy has been extended to study other types of quantum correlation \\cite{bera2018witnessing,datta2018sharing,saha2019sharing}. It has shown that the maximum number of Alices who can simultaneously share steering with a single Bob can also beat the steering monogamy limits \\cite{sasmal2018steering,shenoy2019unbounded,choi2020demonstration}.\n\nHowever, all the steering sharing scenarios \\cite{sasmal2018steering,choi2020demonstration} investigated till now have the following commonalities: the initial shared state is restricted to be the maximum entangled state, the sequential unsharp measurement is only adopted by one side, and the number of measurement settings is not more than 3. Thus some interesting questions raise: whether or not the steering correlation can be kept when the shared state is not pure any more? If there exist multiple observers on both sides, can multiple Alices steer multiple Bobs simultaneously? Compared to the single Bob case, do multiple Bobs make a difference? And how many observers can share steering simultaneously if the number of measurement settings increases? \n\nIn this work, we consider a more general sequential steering scenario featuring that unsharp measurements are sequentially performed on both sides. We investigate how many pairs of Alice and Bob can sequentially demonstrate steering in the above scenario when each party performs $N$-setting equal sharpness measurements. With the $N$-setting linear steering criterion \\cite{cavalcanti2009experimental}, we find no more than 5 Alices can steer a single Bob for a Werner state when $N$ increases from 2 to 16. Then we show how such sequential steering sharing scenarios tolerate the environmental noise and experimental imperfections by analyzing the useful sharpness measurement range of each observer and the minimum purity bound of the initial state. Furthermore, we explore the case when multiple Bobs involved in, reporting a counter-intuitive result that at most 2 Alices can steer 2 Bobs even the number of steering sharing observers larger than 4 in the single Bob case. Finally, we show that our scenario can be used to simulate quantum decoherence channels to effectively change the ability and direction of quantum steering. Our results not only reveal the rich structure of steering sharing but also can be applied to more general scenarios involving high dimension or genuine multipartite quantum steering \\cite{he2013genuine}.\n\n\\section{The both-sides sequential steering sharing scenario}\n\\begin{figure}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[width=10cm]{Fig_1.pdf}\n\t\t\\caption{The scenario of steering sharing with multiple observers on both sides. A two-qubit entangled state is initially shared between a sequence of Alices and Bobs. Multiple Alices implement unsharp measurements on one part of the state successively, and multiple Bobs do similar operations on the other part.}\n\t\t\\label{Fig 1}\n\t\\end{center}\n\\end{figure}\nA schematic of steering sharing scenario with both sides sequential unsharp measurements is shown in Fig.~\\ref{Fig 1}. A pair of two-qubit entangled state $\\rho _{AB} $ is sent to multiple pairs of spatially separated observers. One of the qubits is accessed by $m$ Alices, say, $A_{1}$, $A_{2}$,..., $A_{m}$, while the other qubit is possessed by $n$ Bobs, say, $B_{1} $, $B_{2} $,..., $B_{n}$.\nTo demonstrate the steering between multiple Alices and Bobs at the same time, all observers except the last Alice and Bob should perform unsharp measurements, otherwise, the steerability will be completely destroyed. For convenience, we assume that the sharpness of the $N$-setting measurements that each observer used is equal, which is denoted as $\\lambda_{i}$ and $\\eta_{p}$ for the $ i $-th Alice and the $ p $-th Bob respectively. Thus their corresponding $N$-setting measurements can be represented by $\\lbrace \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{1}}, \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{2}}$,..., $ \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{N}}\\rbrace $ and $\\lbrace \\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{1}}, \\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{2}}$,..., $ \\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{N}}\\rbrace $, where $\\vec{m}_{k}$ and $\\vec{n}_{k}$ represent the measurement directions with $k\\in\\lbrace 1 ,...,\\, N \\rbrace $, $i\\in\\lbrace 1 ,...,\\, m \\rbrace $, $p\\in\\lbrace 1 ,..., \\, n \\rbrace $, $\\lambda_{i} \\in [0,1]$ and $\\eta_{p} \\in [0,1]$. $\\lambda_{i} (\\eta_{p})\\!=\\!0$ corresponds to no measurement, $\\lambda_{i} (\\eta_{p})\\!=\\!1$ implies the measurement is sharp, and $\\lambda_{i} (\\eta_{p})\\in(0,1)$ means it is unsharp. It has been demonstrated that an unsharp measurement is optimal when quality factor $ F $ and the precision $ G $ of the measurement satisfy the trade-off relation $ F^{2}+G^{2}=1 $ \\cite{silva2015multiple}.\nHere, each observer adopts the optimal measurement strategy.\n\nIn the first step, suppose $A_{1}$ wants to convince $B_{1}$ that she can remotely affect his state through local measurements. However, $B_{1}$ does not trust her, so he asks $A_{1}$ to perform a measurement along $\\vec{m}_{k}$. After each run of experiment, $A_{1}$ sends $B_{1}$ the corresponding outcome $ a_{k}\\in \\lbrace 0,1 \\rbrace$ and sends $A_{2}$ the post-measurement state, which can be described by the L\\\"uders ruler \\cite{busch1986unsharp}\n\\begin{center}\n\t\\begin{equation}\n\t\\rho_{AB}\\rightarrow(\\!K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B} )\\rho_{AB}(K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B}),\n\t\\end{equation}\n\\end{center}\nwhere $K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}} K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}}\\!=\\!\\hat{\\Pi}^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}}\\!=\\!(I_{A}+(-1)^{a_{k}} \\lambda_{1}\\,\\vec{m}_{k}\\cdot\\vec{\\sigma})\/2$, $\\vec{\\sigma}\\!=\\!\\{\\sigma_x, \\sigma_y, \\sigma_z\\}$ is the Pauli matrix, and $I_{A}$ $(I_{B})$ is the identity matrix. Repeating the process many times, when $A_{1}$ finishes all the $N$-setting measurements, $B_{1}$ will obtain $2N$ conditional states. Then $B_{1}$ performs some measurements along $\\lbrace \\vec{n}_{1}, \\vec{n}_{2},...,\\vec{n}_{N} \\rbrace$ to analyze whether these conditional states can be described by a local hidden variable state (LHS) model. If they can not, $B_{1}$ is convinced that $A_{1}$ can steer his state, and vice versa. Here, we certify the steering sharing by violating the most widely used $N$-setting linear steering inequality \\cite{cavalcanti2009experimental}, which is defined as $S_N^{1,1} \\leq C_N$ for $A_{1}$ and $B_{1}$, where $S_N^{1,1}\\equiv\\frac{1}{N}\\sum_{k=1}^N\\langle \\hat{\\Pi}^{\\lambda_{1}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{1}}_{\\vec{n}_{k}}\\rangle=\\mathrm{Tr} [\\rho_{AB}\\,( \\hat{\\Pi}^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}}\\otimes \\hat{\\Lambda}^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}})] $, $b_k$ is $B_{1}$'s measurement result, $C_N$ is the maximum value of $S_N$, which can have if the LHS model exists. On the other hand, as $A_{1}$ is assumed to act independently, thus the state shared between $A_{2}$ and $B_{1}$ should be averaged over $A_{1}$'s outputs, i.e.,\n\\begin{equation}\n\\rho^{2,1}_{N}\\!=\\!\\sum_{k=1}^{N} \\sum_{a_{k}=0}^{1}(\\!K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B} )\\rho_{AB}(K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B}).\n\\end{equation}\nSimilarly, the state shared between $A_{1}$ and $B_{2}$ can be described by\n\\begin{equation}\n\\rho^{1,2}_{N}\\!=\\!\\sum_{k=1}^{N} \\sum_{b_{k}=0}^{1}(\\! I_{A} \\otimes K^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}})\\rho_{AB}(\\! I_{A}\\otimes K^{\\eta_{1}\\dagger}_{b_{k}\\mid\\vec{n}_{k}}).\n\\end{equation}\n\nSuppose $B_{1}$ wants to show steering with $A_{2}$ in the next step. They can verify it by calculating the steering parameter\n$S_N^{2,1}\\!\\equiv\\!\\frac{1}{N}\\sum_{k=1}^N\\langle \\hat{\\Pi}^{\\lambda_{2}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{1}}_{\\vec{n}_{k}}\\rangle\\!=\\!\\mathrm{Tr} [\\rho^{2,1}\\,( \\hat{\\Pi}^{\\lambda_{2}}_{a_{k}\\mid\\vec{m}_{k}}\\otimes \\hat{\\Lambda}^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}})]$. If it is larger than $C_N$, then $B_{1}$ succeeds; otherwise, he fails. Considering the first pair Alice and Bob ($A_{1}$ and $B_{1}$) have implemented the matched measurement (when $A_{1}$ performs a measurement along $\\vec{m}_{k}$, and $B_{1}$ should measure along the $\\vec{n}_{k}$), the average state shared between $A_{2}$ and $B_{2}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n\\rho^{2,2}_{N}=\\!\\sum_{k=1}^{N} \\sum_{a_{k}=0 \\atop b_{k}=0}^{1}\\!(K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}}\\!\\otimes\\!K^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}}) \\rho_{AB}(K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}}\\!\\otimes\\!K^{\\eta_{1}\\dagger}_{b_{k}\\mid\\vec{n}_{k}}).\n\\end{split}\n\\end{equation}\n\nActing in analogy with the above process, at any step the state $ \\rho^{i,p}_{N} $ shared between the $ i $-th Alice and the $ p $-th Bob can be obtained by averaging over the previous observers' measurements with the help of the L\\\"uders transformation rule. The corresponding steering criterion can be written as\n\\begin{equation}\\label{eq:SAB}\nS_N^{i,p}\\equiv\\frac{1}{N}\\sum_{k=1}^N\\langle \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{k}}\\rangle\\leq C_N,\\\\\t\n\\end{equation}\nwhere $ \\langle \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{k}}\\rangle=\\mathrm{Tr}[\\rho^{i,p }\\,( \\hat{\\Pi}^{\\lambda_{i}}_{a_{k}\\mid\\vec{m}_{k}}\\otimes \\hat{\\Lambda}^{\\eta_{p}}_{b_{k}\\mid\\vec{n}_{k}})]$. Thus we can investigate the behavior of quantum steering under sequential measurements by comparing the steering parameter with the classical bound in the same measurement setting.\n\n\\section{Sharing the steering of an initial Werner state}\n\\label{Sharing the steering}\nNoting that the environmental effects may turn a pure state into a mixed one and considering the imperfection of the experimental device, one can not prepare a maximum entangled pure state. Here, we take Werner state, the best-known class of mixed entangled states, as an example to investigate the steering sharing among multiple Alices and Bobs with the aid of steering criterion shown in Eq.~(\\ref{eq:SAB}). For qubits, the Werner state is given by \\cite{werner1989quantum}\n\\begin{equation}\n\\rho\\left(\\mu\\right)=\\mu\\vert\\psi\\rangle\n\\langle\\psi\\vert+(1-\\mu)\n\\frac{I}{4},\n\\label{eq:Werner}\n\\end{equation}\nwhere $\\vert\\psi\\rangle\\!=\\!\\frac{1}{\\sqrt{2}}(\\vert01\\rangle-\\vert10\\rangle)$ is the singlet state, $I$ is the identity matrix and $\\mu\\in[0,1]$.\n\nAccording to the symmetrical property of the state, it has been demonstrated that the optimal measurement settings for any Alice and Bob are defined by the directions through antipodal pairs of vertices of a regular polyhedron \\cite{saunders2010experimental}. Thus, we can get 2, 3, 4, 6, 10 measurement settings from the square, octahedron, cube, icosahedron, and dodecahedron, respectively. And it can be further increased by combining the measurement directions from above five regular polyhedrons. In combination with the measurement directions of the icosahedron and dodecahedron, the 16 measurement settings can be obtained \\cite{bennet2012arbitrarily}.\n\nFor the case of multiple Alices and a single Bob, the state sharing among the $ i $-th Alice and the single Bob in the case of $N=2$ settings becomes \n\\begin{equation}\\label{eq:symmetry state for two-settings}\n\\rho_2^{i,1}=\\left( \\begin{array}{cccc}\\frac{1-x}{4} & 0 & 0 & \\frac{-x+z}{4}\\\\\n0 & \\frac{1+x}{4} & -\\frac{x+z}{4} & 0\\\\\n0 & -\\frac{x+z}{4} & \\frac{1+x}{4} & 0\\\\\n\\frac{-x+z}{4} & 0 & 0 & \\frac{1-x}{4}\\end{array} \\right),\n\\end{equation}\nwhere $x\\!=\\!\\frac{1}{2^{i\\!\\!-\\!1}}\\mu\\!\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+\\!\\sqrt{1\\!-\\!{\\lambda_j}^2})\\in[0,1]$ and $z\\!=\\!\\mu\\prod\\limits_{1 \\leq j\\leq i\\!-\\!1}(\\sqrt{1\\!-\\!{\\lambda_j}^2})\\in[0,1]$, $ i\\in\\lbrace 1,2,...,m\\rbrace $. The $j$ is positive integer, and its minimum value is 1.\nWhile the shared state for $N\\geq3$ settings keeps the Werner state's form \n\\begin{equation}\\label{eq:Werner state for three-settings}\n\\rho_N^{i,1}=\\mu'\\vert\\psi\\rangle\n\\langle\\psi\\vert+(1-\\mu')\n\\frac{I}{4},\n\\end{equation}\nwhere $\\mu'\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+2\\!\\sqrt{1\\!-\\!{\\lambda_j}^2})$ $\\in[0,1]$. Obviously, the shared state of each step remains symmetrical, thus the $N$-setting steering inequality Eq. (\\ref{eq:SAB}) is a sufficient and necessary criterion, which can be rewritten as\n\\begin{equation}\\label{S2AB}\nS_2^{i, 1}\\!=\\!\\frac{1}{2^{i\\!-\\!1}}\\mu\\lambda_i\\eta_1\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+\\!F_{\\lambda_j}),\n\\end{equation}\nfor $N\\!=\\!2$ settings and\n\\begin{equation}\\label{SNAB}\nS_N^{i,1}\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\lambda_i\\eta_1\\!\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+2F_{\\lambda_j})\\!,\n\\end{equation}\t\nfor $N\\!\\geq\\!3$ settings. $F_{\\lambda_j}$=$\\sqrt{1\\!-\\!{\\lambda_j}^2}$ represents the quality factors of related measurements. Similarly, the case of a single Alice and multiple Bobs can also be calculated.\n\nFor the case of multiple Alices and Bobs, considering the previous pair of observers adopting the matched measurements (if the Alice performs a measurement along $\\vec{m}_{k}$, the Bob should measure along the $\\vec{n}_{k}$) to verify their state's steering ability, here we choose the optimal method to calculate the state shared by the $ i $-th Alice and the $ p $-th Bob, which can maximize the value of the steering parameter. We take the case that $ i\\geq p>1$ for example, the shared state among the current $ i $-th Alice and $ p $-th Bob for $N\\!=\\!2$ settings is same as the form of Eq. (\\ref{eq:symmetry state for two-settings}), while the value of $x, z$ change to $\\!\\frac{1}{2^{i\\!-\\!1}}\\mu\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\prod\\limits_{1\\leq l \\leq i\\!\\!-p}(1\\!+\\!F_{\\lambda_l})$$\\in[0,1]$, $\\!\\mu\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j}\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}F_{\\lambda_l}$$\\in[0,1]$ respectively. Then the steering parameter can be written as\n\\begin{equation}\\label{S2ABs} \nS_2^{i,p}\\!=\\!\\frac{1}{2^{i\\!-\\!1}}\\mu\\lambda_i\\eta_p\\!\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\!\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}(1\\!+\\!F_{\\lambda_l}), \n\\end{equation} \nwhere the $l$ is positive integer.\n\nWhen $N\\!\\geq\\!3$, their shared state still follows the Werner state's form of Eq. (\\ref{eq:Werner state for three-settings}), where $\\mu'\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!2\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}(1\\!+\\!2\\!F_{\\lambda_l})$ $\\in[0,1]$. And the steering parameter becomes\n\\begin{equation}\\label{SNABs} \nS_N^{i,p}\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\lambda_i\\eta_p\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!2\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}(1\\!+\\!2\\!F_{\\lambda_l}).\n\\end{equation}\nThe other case that $ p\\geq i>1$ can be obtained with the same method.\n\nObviously, the unsharp measurement strategy used here is optimal. For Werner state, the classical bound $C_N\\!=\\!\\lbrace1\/\\sqrt{2},\\,1\/\\sqrt{3},\\,1\/\\sqrt{3},\\,0.5393,\\,0.5236,\\,0.503,\\,0.5\\rbrace$ when $N\\!=\\!\\lbrace 2,3,4,6,10,16,\\infty \\rbrace$ respectively \\cite{saunders2010experimental,pramanik2019nonlocal}. \nSince the classical bound of $N\\!=\\!16$ is very close to that of infinite measurement settings, we implement $N\\!=\\!\\lbrace 2,3,4,6,10,16 \\rbrace$ to investigate the behavior of quantum steering in this work.\n\t\n\\subsection{Multiple Alices and a single Bob}\n\\begin{figure}[H]\n\t\\begin{center}\n\t\t\\includegraphics[width=9cm, height=6cm]{Fig_2.pdf}\n\t\t\\caption{The relationship between the maximum number of Alices $N_A^\\mathrm{max}$ who can share steering with a single Bob and the number of measurement settings $N$.}\n\t\t\\label{Fig 2}\t\n\t\\end{center}\n\\end{figure}\n\\begin{table}[]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{The range of different measurement sharpness, $\\lambda_1$, $\\lambda_2$, $\\lambda_3$ and $\\lambda_4$, when steering sharing is achieved for the pure initial shared state ($\\mu=1$), where $N$ is the number of measurement settings, and $N_A$, $N_B$ represent the number of Alices and Bob respectively. $\\lambda_1$, $\\lambda_2$, $\\lambda_3$ and $\\lambda_4$ denote the measurement sharpness of $A_1$, $A_2$, $A_3$ and $A_4$ respectively, and similarly, $\\eta_1$ is the sharpness of $B_1$'s measurement. $\\mu_\\mathrm{min}$ indicates the purity infimum of the initial Werner state, which reflects the noise robustness. As long as $\\mu$ is greater than $\\mu_\\mathrm{min}$, steering sharing happened in the case $\\mu\\!=\\!1$ can still be realized.}\\label{Table 1}\n \\centering \n \\scriptsize\n\t\\begin{tabular}{|p{4pt}<{\\centering}|p{3pt}<{\\centering}|p{3pt}<{\\centering}|p{51pt}<{\\centering}|p{51pt}<{\\centering}|p{51pt}<{\\centering}|p{10pt}<{\\centering}|p{30pt}<{\\centering}|p{20pt}<{\\centering}|}\n\t\t\\hline\t\t\n\t\t$N$&$N_A$&$N_B$&$\\lambda_1$&$\\lambda_2$&$\\lambda_3$& $\\lambda_4$&$\\eta_1$&$\\mu_\\mathrm{min}$ \\\\\n\t\t\\hline\n\t\t2&1&1&$[0.707(2),1]$&--&--&--&$[0.707(2),1]$&$0.707(2)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t2&2&1&$[0.707(2),0.910(1)]$&1&--&--&$[0.884(1),1]$&$0.891(9)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t3\/4&1&1&$[0.577(4),1]$&--&--&--&$[0.577(4),1]$&$0.577(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t3\/4&2&1& $[0.577(4),0.930(6)]$&1&--&--&$[0.756(0),1]$&$0.759(8)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t3\/4&3&1&$[0.577(4),0.773(3)]$&$[0.657(9),0.873(5)]$&1&--&$1$&$0.909(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&1&1&$[0.539(4),1]$&--&--&--&$[0.539(4),1]$&$0.539(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&2&1&$[0.539(4),0.951(0)]$&1&--&--&$[0.706(2),1]$&$0.706(7)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&3&1&$[0.539(4),0.828(9)]$&$[0.602(8),0.914(7)]$&1&--&1&$0.846(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&4&1&$[0.539(4),0.643(7)]$&$[0.602(8),0.710(2)]$&$[0.707(5),0.829(1)]$&1&1&$0.965(5)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&1&1&$[0.523(7),1]$&--& --&--&$[0.523(7),1]$&$0.523(7)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&2&1&$[0.523(7),0.958(4)]$&1&--&--&$[0.685(7),1]$&$0.686(1)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&3&1&$[0.523(7),0.848(9)]$&$[0.581(0),0.928(4)]$&1&--&1&$0.816(0)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&4&1&$[0.523(7),0.677(5)]$&$[0.581(0),0.749(6)]$&$[0.674(3),0.859(3)]$&1&1&$0.930(2)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&1&1&$[0.503(1),1]$&--&--&--&$[0.503(1),1]$&$0.503(1)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&2&1&$[0.503(1),0.967(0)]$&1&--&--&$[0.658(7),1]$&$0.658(7)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&3&1& $[0.503(1),0.872(8)]$&$[0.553(1),0.944(1)]$&1&--&1&$0.783(3)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&4&1& $[0.503(1),0.727(0)]$&$[0.553(1),0.795(4)]$&$[0.626(3),0.898(3)]$&1&1&$0.888(9)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&5&1&$[0.503(1),0.545(4)]$&$[0.553(1),0.612(5)]$&$[0.626(6),0.679(6)]$&0.766&1&$0.979(5)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t\\hline\n\t\\end{tabular}\n\t\\normalsize\n\\end{table} \nFirstly, we take multiple Alices and a single Bob as an example to explore how many observers in one part can simultaneously steer the state of a single observer in the other part in different settings. It is obvious that the steering parameter increases with the increasing of current measurement sharpness and the decreasing of previous measurement sharpness. Since the previous measurement may decrease the steerability of the current shared state, the measurement sharpness of the latter Alices would be increased to obtain enough information to show their steerability, i.e., $\\lambda_1<\\lambda_2<$,...,$<\\lambda_m$. And the steering sharing process can continue, with each latter Alice and Bob being able to violate steering inequality with the average shared state obtained from the previous stage, as long as $\\lambda_i\\!<\\!1$ and $\\eta_1\\!<\\!1$. From this condition, one can obtain the maximum number of Alices $N_A^\\mathrm{max}$ who can share steering simultaneously with a single Bob. The result is presented in Fig.~\\ref{Fig 2}. It is obvious that as the number of measurement settings $N$ increases, the overall tendency of $N_A^\\mathrm{max}$ rises. We find at most 5 Alices can simultaneously steer Bob's state when the number of measurement setting reaches 16. Interestingly, for some special case $N_A^\\mathrm{max}$ remains the same even if $N$ increases (such as $N\\!=\\!3,4$, or $N\\!=\\!6,10$). Note that it was conjectured in Ref.~\\cite{sasmal2018steering} that at most $N$ Alices can exhibit steering with a single Bob by the violation of $N$-setting linear inequality. From our results, it seems that this conjecture is not true. \n\nWe further calculate the useful sharpness parameter regions for all possible sharing scenarios with the maximally entangled initial state. The results are summarized in Table~\\ref{Table 1}. Here, we assume Bob performs sharp measurements when $N_A\\geq 3$ and the final Alice also performs sharp measurements when $N_A\\geq 2$, while in other cases, the observer's measurements are unsharp. It clearly indicates that the useful measurement sharpness interval of these observers decreases with the number of Alices increasing and the number of the measurement settings decreasing. For the case of 2 Alices and a single Bob, we find the ranges of the first Alice's sharpness $\\lambda_1$ and the first Bob's sharpness $\\eta_1$ are respectively expanded about 2.3 and 2.9 times, as the number of measurement settings increases from 2 to 16, making it easier to apply directly to experiments.\n\nIt should be noted that all the above results are restricted to a pure initial shared state. Considering the decoherence effect of environmental noise and the imperfection of the experimental device, we further investigate whether or not the steering correlation can be kept when the shared state is not pure any more. We find that it can be kept indeed. The minimum purity bound of the initial state $\\mu_\\mathrm{min}$ is presented in the last column of Table \\ref{Table 1}. Obviously, for any possible steering sharing scenarios, there exist a finite continuous range of purity such that these Alices can share steering with Bob. And for a fixed number of observers, the more measurement settings, the greater the purity range and the stronger the robustness.\n\n\\subsection{Multiple Alices and Bobs}\n\\begin{figure}[H]\n\t\\begin{center}\n\t\t\n\t\t\\subfigure[ ]{\\includegraphics[width=3.75cm]{Fig_3_a.pdf}}\\\\\n\t\t\\subfigure[ ]{\\includegraphics[width=5.5cm]{Fig_3_b.pdf}}\n\t\t\\subfigure[ ]{\\includegraphics[width=5.5cm]{Fig_3_c.pdf}}\n\t\t\\caption{(a) The schematic diagram for the 2 Alices and 2 Bobs is displayed via 3 or more settings measurement. Here, the arrow indicates the steering direction. (b) Steering parameters $S_3^{i,p}$ ($i\\!=\\!1,2$, and $p\\!=\\!1,2$) are presented for 3-setting measurements as a function of $\\lambda_{1}$ and $\\eta_{1}$. $\\lambda_{2}=1$ and $\\eta_{2}=1$ indicates that $A_2$ and $B_2$ implement the sharp measurements. The green, blue, red, and purple lines correspond to $S_3^{1,1}\\!=\\!C_3$, $S_3^{1,2}\\!=\\!C_3$, $S_3^{2,1}\\!=\\!C_3$, and $S_3^{2,2}\\!=\\!C_3$ respectively. The regions where the corresponding colored arrows point to indicate that the $S_3^{1,1}$, $S_3^{1,2}$, $S_3^{2,1}$, and $S_3^{2,2}$ exceed $C_3$, respectively. (c) Displaying the steering parameters $S_{16}^{i,p}$ ($i\\!=\\!1,2$, and $p\\!=\\!1,2$) for 16-setting measurements versus $\\lambda_{1}$ and $\\eta_{1}$. Similarly, The regions where the corresponding colored arrows point to mean that the violation of linear inequality between $A_1$-$B_1$, $A_1$-$B_2$, $A_2$-$B_1$, $A_2$-$B_2$, respectively. The overlapping regions in (b) and (c) colored in yellow demonstrate the steering sharing among 2 Alices and 2 Bobs.}\n\t\t\\label{Fig 3}\n\t\\end{center}\n\\end{figure}\t\nIn the previous section, we get the number limitation of Alice who can demonstrate steering with a single Bob. Now, we address the question of whether these Alices can further share steering with more Bobs. We find that it is not possible to increase the number of Bobs when the number of Alices reaches the maximum value in the single Bob scenario. \nHowever, we can reduce the number of observers on one side to increase that on the other side, and then make it possible that multiple Alices show steering with multiple Bobs. Counterintuitively, we find that at most 2 Alices can be simultaneously steered by 2 Bobs in the multiple Alices and Bobs scenario even if the total number of steering shared observers in the single Bob scenario is greater than 4 (see Appendix for more details).\n\nTaking the first two Alices and Bobs as an example, the 2 Alices and 2 Bobs successful steering sharing scenario is depicted Fig. \\ref{Fig 3}(a) and the relationship between the steering parameters and sharpness parameters in the case of 3-setting measurements and 16-setting measurements is presented Fig. \\ref{Fig 3}(b) and Fig. \\ref{Fig 3}(c). The yellow region represents the valid ranges of $\\lambda_{1}$ and $\\eta_{1}$ where 2 Alices and 2 Bobs share steering at the same time. Due to the symmetrical property of the state, $\\lambda_{1}$ and $\\eta_{1}$ have the same ranges. For the 3-setting measurements, they both are $[0.756(1),0.802(5)]$ which is much smaller than the valid ranges of $\\lambda_{1}$ and $\\lambda_{2}$ in the case of 3 Alices and a single Bob with the same measurement settings, indicating it is harder to sharing quantum steering between multiple Alices and Bobs. However, one can improve its robustness by adding the measurement settings. Fig. \\ref{Fig 3}(b) and Fig. \\ref{Fig 3}(c) show that the useful ranges of $\\lambda_{1}$ and $\\lambda_{2}$ can be expanded more than 10 times as the measurement settings increase from 3 to 16. \n\n\\section{Application}\n\\label{Application}\nNote that if the disturbance caused by the former observer's measurement is regarded as noise, our steering sharing protocol can also be applied to investigate the dynamic of steering in the presence of decoherence \\cite{pramanik2019nonlocal}, such as steering sudden death and revival \\cite{sun2017recovering}. Especially, our 3 settings unsharp measurement strategy is essentially equivalent to the depolarizing channel. By changing the former observer's measurement sharpness, the steering ability and direction of the current observers can be controlled. For example, in our 2 Alices and 2 Bobs steering sharing scenario, $A_2$ and $B_2$ can share steering if $\\lbrace \\lambda_{1}, \\eta_{1} \\rbrace$ locates in left side of purple line ($S_3^{2,2}=C_3$ ) in Fig. \\ref{Fig 3}(b), otherwise they cannot. If the initial Werner state is replaced by an asymmetric state \\cite{xiao2017demonstration} or $A_1$ and $B_1$ adopt some asymmetric measurements, a tunable $\\lbrace \\lambda_{1}, \\eta_{1} \\rbrace$ further allows $A_2$ and $B_2$ to exhibit their steerabilities from both directions to only one direction. \n\n\\section{Conclusion and Discussion}\n\\label{Conclusion and Discussion}\nIn this work, we discuss a new steering sharing scenario, where half of an entangled pair is accessed by a sequence of Alices, and the other half is distributed to multiple Bobs. We address the question of how many pairs of Alice and Bob can demonstrate quantum steering by violating the $ N $-setting steering inequality where $N=2,3,4,6,10,16$. Contrary to the conjectured proposed by Sasmal \\textit{et al.} \\cite{sasmal2018steering}, we find at most 5 Alices can steer a single Bob and no more than 2 observers can be steered in the multiple Alices and Bobs scenario when the sharpness of the $N$-setting measurements that each Alice and Bob used is equal. We also provide the useful sharpness parameter ranges and the minimum purity of the initial state for different steering sharing cases and give evidence that they increase as the number of observers decreases and the number of measurement settings increases. The noise robustness of our sharing scenario makes our results applicable to the experimental demonstration. On the other hand, we show that our protocol can also be applied to investigate the dynamic of steering in a noise channel and even control the steering direction.\n\nThe shareable steering is a primary resource for some practical and commercial quantum information processing tasks where the general consumers may not want to trust their providers, such as, in the context of quantum internet \\cite{Kimble2008internet}, secret sharing \\cite{Armstrong2015secret}, and random number generation \\cite{Cavalcanti2015random}. It is thus of importance to further increase the shareable observers to utilize it for many times which could be realized by adopting multipartite entangled states \\cite{gupta2021genuine} or allow the sequential observers in the above scenario to share some classical information. We will carry out some researches in these directions in the near future.\n\n\\begin{acknowledgements}\t\t\nThis work was supported by the National Natural Science Foundation Regional Innovation and Development Joint Fund (Grant No. 932021070), the National Natural Science Foundation of China (Grant No. 912122020), the China Postdoctoral Science Foundation (Grant No. 861905020051), the Fundamental Research Funds for the Central Universities (Grants No. 841912027, and 842041012), the Applied Research Project of Postdoctoral Fellows in Qingdao (Grant No. 861905040045), and the Young Talents Project at Ocean University of China (Grant No. 861901013107). The authors thank Jie Zhu for fruitful discussions.\n\\end{acknowledgements}\n\\newpage\n\\section{Appendix steering sharing with three Alices and two Bobs}\n\\begin{figure}[H]\n\t\\begin{center}\n\t\n\t\t\\subfigure[ ]{\\includegraphics[width=5.2cm]{Fig_4_a.pdf}}\n\t\t\\subfigure[ ]{\\includegraphics[width=6.3cm]{Fig_4_b.pdf}}\n\t\t\\caption{(a) The schematic diagram of steering sharing with 3 Alices and 2 Bobs. The arrow indicates the steering direction, and the lines of the same type (solid or dotted) indicate that the steering can demonstrate simultaneously, and vice versa. (b) The region of the violation of 16-setting steering inequality for the maximum entangled state ($\\mu\\!=1$) with $\\lambda_{3}=1$ and $\\eta_{2}=1$. The yellow region and dark purple area represent that the three Alices can steer the state of the first Bob and the second Bob, respectively. There is no overlap indicates the first three Alices can not share steering with the first two Bob at the same time.}\n\t\t\\label{Fig 4}\n\t\\end{center}\n\\end{figure}\nIn the third part of the main text, we explore the steering sharing scenario with multiple Alices and Bobs. In this case, we find that only 2 Alices can detect steering with 2 Bobs at the same time. Here, we take the first three Alices ($A_1,A_2,A_3 $) and the first two Bobs ($B_1,B_2 $) as an example to illustrate why it is impossible to further increase the number of observers as the number of measurement settings increases to 16. As shown in Fig. \\ref{Fig 4}(b), the yellow region represents the case of $A_1,A_2,A_3 $ can steer the $B_1$'s state. The dark purple region represents the case of $A_1,A_2,A_3 $ can steer the $B_2$'s state. It clearly shows that these Alices and Bobs can not share steering at the same time, because there is no overlap in the two regions even though they initially share a maximum entangled state ($\\mu\\!=\\!1$) and the last observer at each side performs sharp measurements ($\\lambda_{3}\\!=\\!1$ and $\\eta_{2}\\!=\\!1$). Thus, for other fewer measurement settings or mixed initial state, it certainly doesn't exist steering sharing among 3 Alices and 2 Bobs.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection{Introduction}\nThe astonishing experimental accomplishments in the optical control and manipulation of cold atomic gases in the past decade now allow to deterministically load them in extremely well controlled optical traps of almost any shape down to effectively zero temperature. A particular fruitful example are periodic optical lattices in which many intriguing phenomena of solid state physics can be studied with unprecedented control\\cite{bloch2008many}.\n\nIn contrast to conventional solids, however, the spatial order and lattice geometry is fixed by the external lasers and does not appear from a self consistent dynamics of particle interactions. As the back-action of the atoms onto the confining light fields is generally negligible\\cite{asboth2008optomechanical}, local lattice perturbations do not propagate and long range interactions, which appear as phonons in solids are absent. This changes for spatially tightly confined fields as in small optical resonators or optical micro-structures\\cite{domokos2002quantum}. \n\nExperimentally it is very challenging to implement and load microtraps close to optical microstructures, where such atom field coupling is enhanced\\cite{colombe2007strong, alton2010strong, schleier2011optomechanical}. In an important step Rauschenbeutel and coworkers, however, recently managed to trap atoms in an array of optical dipole traps generated by two color evanescent light fields alongside a tapered optical fiber\\cite{vetsch2010optical} where the backaction of even a single atom on the propagating fiber field is surprisingly strong\\cite{domokos2002quantum,chang2012cavity}. This setup was improved with higher control and coupling by other groups recently\\cite{goban2012demonstration}. With the atoms firmly trapped within the evanescent field of the fiber modes, field mediated atom-atom interaction and collective coupling to the light modes play a decisive role in this a setup\\cite{zoubi2010hybrid}.\n\nAlready a decade ago it was theoretically predicted\\cite{domokos2002collective} and experimentally confirmed\\cite{black2003observation,baumann2010dicke,arnold2012self} that light scattering within optical resonators induces self-ordering of atoms in regular patterns maximizing collective coupling to the cavity mode\\cite{ritsch2012cold,gopalakrishnan2009emergent}. This transition can be directly monitored from the super-radiant light scattering \\cite{black2003observation,arnold2012self} and appears also at zero temperature in the quantum regime as phase transition from a superfluid to a supersolid\\cite{baumann2010dicke}. In an recent proposal Chang and coworkers predicted, that nanofiber-mediated infinite range dipole-dipole coupling can also induce stable regular patterns of laser illuminated atoms trapped along the fiber\\cite{chang2012self}. The stable configurations, characterized by minimal dipole interaction energy, assume surprising configurations and exhibit characteristic collective light scattering. \n\nHere we develop a generalized many particle model to study the properties of such a light scattering induced crystallization of an laser illuminated ultracold gas in an elongated 1D trap at finite temperature as schematically drawn in Fig.\\ref{fig1}. The atoms trapped parallel to the fiber are illuminated at right angle by a Gaussian laser beam of frequency $\\omega_L$, sufficiently detuned from any atomic resonance so that spontaneous emission plays only a minor role and their polarizability $\\alpha$ has only a negligible imaginary part.\n\n\\begin{figure}[h!]\n\n\\includegraphics[width=8cm]{Bild1.jpg}\n\\caption{Cigar-shaped atomic gas alongside optical nanoguide}\\label{fig1}\n\\end{figure}\nThe laser field $\\mathbf E_L=(\\mathcal E_L(\\mathbf x)e^{-i\\omega_Lt}+c.c.)\\mathbf e_L$ gives rise to the field $\\mathbf E_s$ scattered by the atoms. Neglecting polarization effects and retardation, it can be written as $\\mathbf E_s=(\\mathcal E_s(\\mathbf x,t)e^{-i\\omega_Lt}+c.c.)\\mathbf e_L$, its envelope satisfying Helmholtz's equation\n\\begin{subequations}\\label{VlasovHelmholtz}\n\\eq{\\label{FullHelmholtz}\\nabla^2\\mathcal{E}_s +k_L^2\\big(n_F^2+\\chi\\big)\\mathcal{E}_s=-k_L^2\\chi\\mathcal{E}_L.}\nHere, $n_F(\\mathbf x)$ denotes the refractive index profile of the optical fiber and $\\chi(\\mathbf x,t) ={\\alpha}\\rho(\\mathbf x,t)\/{\\epsilon_0}$ is the susceptibility of the particles. The particle density distribution $\\rho$ is obtained from the momentum integral \n$\\rho(\\mathbf x,t)=\\int F(\\mathbf x,\\mathbf p,t)d^3p$ \nover the one-body distribution function $F(\\mathbf x,\\mathbf p,t)$ proper to the gas. \nIn the classical mean-field limit it satisfies Vlasov's equation\n\\eq{\\label{FullVlasov}\n\\frac{\\partial F}{\\partial t}+\\frac{\\mathbf p}{m}\\!\\cdot\\!\\frac{\\partial F}{\\partial \\mathbf x}-\\frac{\\partial }{\\partial \\mathbf x}\\left(\\Phi_d+U_T\\right)\\!\\cdot\\!\\frac{\\partial F}{\\partial \\mathbf p} =0.}\n\\end{subequations}\nHere, $U_T$ denotes the elongated dipole trap potential and \\eq{\\Phi_d=-\\alpha \\left|\\mathcal{E}_s+\\mathcal E_L\\right|^2} the optical potential due to pump laser and fiber field. For strong radial confinement of the gas the one-body distribution approximately factorizes into a longitudinal and a transverse part,\n$F(\\mathbf x,\\mathbf p,t)\\simeq f(z,p_z,t) F_\\perp(\\mathbf x_\\perp,\\mathbf p_\\perp)$, where $F_\\perp$ is the Maxwell-Boltzmann distribution for the transverse degrees of freedom: $F_\\perp:= Z_\\perp^{-1}\\exp\\left[-\\beta\\left(\\frac{\\mathbf p_\\perp^2}{2m}+U_\\perp(\\mathbf x_\\perp)\\right)\\right]$. $\\beta$ denotes the inverse thermal energy and $U_\\perp$ the radial confining potential.\n\nIn the absence of atoms the fiber supports only a single relevant TE mode, which -- within the framework of scalar theory -- is supposed to possess a propagation constant $\\beta_m$ and a normalized transverse mode function $u(x,y)$ extending outside the fiber \\cite{vetsch2010optical,chang2012self}. The dominant forces on the particles along $z$ are due to photon scattering into and out of the fiber. As long as the total atomic susceptibility stays small even for large particle numbers, the radial fiber mode function is only weakly perturbed and we can set $\\mathcal E_s(\\mathbf x,t)\\simeq\\sqrt{A}\\,E(z,t)u(x,y)$ with the cross section $A:=\\left(\\int d^2x_\\perp d^2p_\\perp u^2 F_\\perp \\right)^{-1}$. Integrating over the transverse degrees of freedom we finally arrive at an effective description of the longitudinal dynamics\n\\begin{subequations}\\label{HV}\n\\eq{\\label{Helmholtz}\\frac{\\partial^2E}{\\partial z^2}+\\left(\\beta_m^2+k_L^2\\tilde\\chi\\right)E=-k_L^2 \\tilde\\chi E_L,}\n\\eq{\\label{Vlasov}\\frac{\\partial f}{\\partial t}+\\frac{p_z}{m}\\frac{\\partial f}{\\partial z}-\\frac{\\partial }{\\partial z}\\!\\left(U-\\alpha\\!\\left[|E|^2+2E_LE_r\\right]\\right)\\!\\frac{\\partial f}{\\partial p_z} =0,}\n\\end{subequations}\nwhere $E_r$ is the real part of $E$. The effective laser field is given by $E_L:=\\sqrt{A}\\,\\int d^2x_\\perp d^2p_\\perp\\mathcal{E}_L u F_\\perp$ and the local atom-fiber coupling is governed by the effective susceptibility\n\\eq{\\label{Susc}\\tilde\\chi(z,t):=\\frac{\\alpha}{\\epsilon_0 A}\\int^{\\infty}_{-\\infty} f(z,p_z,t)dp_z}\nproportional to the atomic line density. To ensure physically consistent solutions, equation \\eqref{Helmholtz} for the electric field has to fulfill Sommerfeld's radiation conditions\n\\begin{equation}\n\\label{BC}\\frac{\\partial E}{\\partial z}=\\pm i\\beta_m E(z,t),~~z\\rightarrow \\pm\\infty,\n\\end{equation}\nwhich ensure outgoing waves at infinity. \n\nEquations \\eqref{HV} can at least numerically be solved directly. At this point we nevertheless restrict ourselves to such stationary solutions, which correspond to a gas in thermal equilibrium\\cite{asboth2005self}, where the particles are distributed according to the Maxwell-Boltzmann distribution\n\\eq{\\label{ThermEq}f(z,p_z)=Z^{-1} e^{-\\beta(p_z^2\/2m+U)}\\,e^{\\beta\\alpha\\left(|E|^2+2E_pE_r\\right)}.} \n\nSubstituting the effective susceptibility \\eqref{Susc} obtained from a thermal distribution \\eqref{ThermEq} into the effective Helmholtz equation \\eqref{Helmholtz} leads to a highly nonlinear equation, which determines the selfconsistent electric field. We assume here that the external part of the potential, $U(z)$, can be harmonically approximated by $U(z)=\\frac{1}{2}m\\omega_z^2 z^2$ within the (very large) thermal extension of the gas $l_z:=2k_\\mathrm{B} T\/m\\omega_z^2\\gg\\beta_m^{-1}$, and we will ignore the dependence of the driving laser amplitude on position, i.e. $E_L(z)=E_L$.\n \nWith these assumptions one sees that the zero field solution $E(z)\\approx 0$ and\n\\eq{\\label{ThermEq0}f(z,p_z)\\equiv f_0(z,p_z)=Z^{-1} e^{-\\beta(p_z^2\/2m+U)}}\nalways solves Helmholtz's equation, which we will call the normal phase.\n\n\\begin{figure}\n\t\\centering\n\t\t \\includegraphics[width=8cm]{Bild3.jpg}\n\t\\caption{Properties of three coexisting selfordered solutions on different branches $n=0$ ($(1a)-(1c)$), $n=1$ ($(2a)-(2c)$) and $n=2$ ($(3a)-(3c)$) in the strong collective coupling regime for $\\zeta_0=150\/\\pi$ and $\\varepsilon=9.5\\varepsilon_c$. The upper plots (a) show the atomic density along the fiber, while the plots labeled by (c) depict the local fraction of right $N_+$ (blue curve) and left $N_-$ (red curve) traveling photons, the number of zeros being equal to the branch number $n$. The plots labeled (b) depict the envelopes of the two parts of the optical potential. Blue indicates the potential from scattering between pump laser and fiber, the brown area shows the part due to photon redistribution within the fiber. For $n=0$ the latter dominates in the center of the cloud causing a density modulation with a period of one half of $\\lambda_m=2\\pi\/\\beta_m$. This region is broken up into three for $n=1$ and into five, for $n=2$.}\\label{Bild2}\n\\end{figure}\nBefore we investigate the possibility of selfordered solutions, let us examine the dynamical stability of the normal phase, using the Helmholtz-Vlasov equations \\eqref{HV}, linearized around the unordered state\n\\begin{subequations}\\label{linVlas}\n\\eq{\\frac{\\partial^2E}{\\partial z^2}+\\beta_m^2E=-E_L\\frac{k_L^2\\alpha}{\\epsilon_0A}\\int^{\\infty}_{-\\infty}f_1(z,p_z,t)dp_z ,}\n\\eq{\\frac{\\partial f_1}{\\partial t}+\\frac{p_z}{m}\\frac{\\partial f_1}{\\partial z}-\\frac{d U}{d z}\\frac{\\partial f_1}{\\partial p_z}=-2\\alpha \\frac{\\partial E_L E_r}{\\partial z}\\frac{\\partial f_0}{\\partial p_z}.}\n\\end{subequations}\nHere, $f_1(z,p_z,t)$ represents a small deviation from the normal phase, such that the complete distribution function is given by $f(z,p_z,t)=f_0(z,p_z)+f_1(z,p_z,t)$.\nWe seek solutions to \\eqref{linVlas}, which are of the form\n\\begin{subequations}\\label{NormalMode}\n\\eq{f_1(z,p_z,t)=\\psi(z,p_z)e^{st}+\\psi^*(z,p_z)e^{s^*t},}\n\\eq{E(z,t)=a(z)e^{st}+b^*(z)e^{s^*t}.}\n\\end{subequations}\nIn this Ansatz, $(\\psi,a,b)$ will be referred to as normal mode and $s=\\gamma+i\\omega$ as the complex valued mode parameter.\nWhenever $\\gamma>0$, the deviation from equilibrium defined by the normal mode increases exponentially in time, causing the eventual destruction of the normal phase. The existence of such a mode implies dynamical instability. With the help of asymptotic expansions, we find that if \\eqref{NormalMode} is to solve \\eqref{linVlas} and satisfy the radiation conditions, the mode parameter must satisfy $D_n(s)=0$ for some $n\\in\\mathbb{Z}$, where\n\\eq{D_n(s)=(2n+1)\\pi+i\\frac{k_L^2\\alpha^2}{\\epsilon_0A}\\iint^{\\infty}_{-\\infty}\\frac{E_L^2\\,\\partial f_0\/\\partial p_z}{s+i\\beta_m v_z} dp_z dz.}\n\nThis condition first demands $\\omega=0$. Introducing the collective coupling parameter $\\zeta_0$ and effective pump strength $\\varepsilon$\n\\eq{\\zeta_0:=\\frac{k_p}{\\beta_m}\\frac{N\\alpha}{A\\lambda_L\\epsilon_0}, \\; \\varepsilon:=\\frac{\\alpha E_L^2}{k_BT},}\nwe then see that there exists at least one normal mode with a positive growth rate $\\gamma>0$, if and only if\n\\eq{\\label{CC1}\\varepsilon>\\frac{1}{2\\zeta_0}=:\\varepsilon_c.}\n\nHence as our first important result we see that beyond this pump threshold \\eqref{CC1} the normal phase ceases to be stable and no longer represents a physical state of the system. We therefore study other solutions of \\eqref{HV} for $\\varepsilon>\\varepsilon_c$.\nFortunately, we can obtain approximate solutions with the help of perturbation theory in the weak collective coupling regime, $\\zeta_0\\leq1$, as well as in the strong collective coupling regime $\\zeta_0\\gg1$. The real and imaginary parts of the electric field envelope $E_r, E_i$ \\eq{E_{r,i}(z)=a_{r,i}(z)\\cos(\\beta_m z+\\phi_{r,i}(z))E_L,} can be assumed to behave almost harmonically with phases $\\phi_r, \\phi_i$ and amplitudes $a_r, a_i$ that vary slowly on a scale defined by $\\beta_m^{-1}$. \nThis amplitudes and phases are directly related to the complex amplitudes of the more familiar decomposition into left and right running waves, $E=(E_+e^{i\\beta_mz}+E_-e^{-i\\beta_mz})E_L$ via $E_\\pm=a_r e^{\\pm i\\phi_r}+ia_i e^{\\pm i\\phi_i}$.\nA perturbative analysis of the steady state Helmholtz equation reveals, that \n$\\Theta:=\\frac{a_r^2+a_i^2}{2}=\\frac{1}{2}(|E_+|^2+|E_-|^2)$, proportional to the sum of locally right going and left going photons, is spatially constant and will thus serve us as order parameter.\n\nLet us first consider the weak collective coupling regime, where scattering from the laser into the fiber and vice versa dominates over scattering within the fiber. The demand to satisfy the radiation conditions \\eqref{BC} leads to the equation determining this order parameter\n\n\\eq{\\label{OP}2\\zeta_0 I_1\\left(2\\varepsilon\\sqrt{\\Theta}\\right)=(1+2n)\\sqrt{\\Theta} I_0\\left(2\\varepsilon\\sqrt{\\Theta}\\right),~~n\\in\\mathbb{N}_0,}\nwhere $I_k$ denotes the $k$th modified Bessel function of the first kind.\n\nAs quite a surprise one observes that the solution is not unique and for\n$\\varepsilon_c<\\varepsilon<\\frac{1+2n}{2\\zeta_0}$,\nthere exist $n$ different solutions ${\\Theta_n}$ as shown in the example of figure \\ref{Bild3}. As the phase of the outgoing scattered light is not determined, each solution of \\eqref{OP} corresponds to an infinite family of solutions of \\eqref{HV} with the periodic density modulation slightly displaced under a slow envelope. The appearance of this degeneracy of course corresponds to breaking of a continuous symmetry. It is important to note that the point of first appearance of an ordered family of solutions, $\\varepsilon=\\varepsilon_c$, exactly coincides with the point where the normal phase becomes unstable. All this is confirmed by a numerical solution of the underlying equations \\eqref{HV}.\nFocusing on the region close to the first branching point, we find the behavior of the order parameter to be given by \\eq{\\Theta\\sim\\varepsilon-\\varepsilon_c,~~\\zeta_0<1}\nand\n\\eq{\\Theta\\sim\\left(\\varepsilon-\\varepsilon_c\\right)^{1\/2},~~\\zeta_0=1.}\nCuriously, if $\\zeta_0>1$, the order parameter is discontinuous, exhibiting a jump of magnitude\n\\eq{\\Delta\\Theta=16(\\zeta_0-1).}\nThe results of the perturbation theory hold only as long as $\\Theta\\ll1$ and the predicted jump soon violates this assumption.\nLet us therefore turn to the strong collective coupling regime. Again, each possible value of the order parameter corresponds to an infinite family of stationary solutions and\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{Bild2.jpg}\n\n\t\\caption{Branches of the order parameter vs energy ratio according to \\eqref{OP} and \\eqref{OP1} for $n=0,\\ldots,3$ in the weak collective coupling regime $(a)$, with $\\zeta_0=0.05$ and in the strong collective coupling regime $(b)$, with $\\zeta_0=75\/\\pi$. Note that in the latter regime all branches converge and there may be even more than $n$ families of solutions for $\\varepsilon_c<\\varepsilon<(1+2n)\\varepsilon_c$.}\\label{Bild3}\n\\end{figure}\nany nonzero value for $\\Theta$ satisfies\n\\eq{\\label{OP1}\\zeta_0 \\varepsilon\\pi=4(1+2n)P(\\Theta)K[P^2(\\Theta)\\Theta^2],~~n\\in\\mathbb{N}_0,}\nwhere $P(\\Theta):=\\frac{1+2\\varepsilon \\Theta}{4+6\\varepsilon \\Theta}$ and $K$ denotes the complete elliptic integral of the first kind.\nIn this regime, there are at least $n$ families of ordered solutions whenever $1<2\\zeta_0\\varepsilon<1+2n$. For an example see figure \\ref{Bild3}.\nFurthermore, we find that according to perturbation theory, the order parameter is bounded by $\\Theta<4$. Figure \\ref{Bild2} depicts some properties of the solutions corresponding to possible values of the order parameter. It should be noted that while we have here shown the existence of multiple families of selfordered thermal solutions of the Vlasov-Helmholtz equations, nothing has been said about their dynamical stability properties, which remains an open problem.\n\\par\nCertain phenomena called optical binding, similar to the ordering discussed in this work, have been observed with laser illuminated small beads in liquids\\cite{burns1990optical} in 1D and even 2D geometries as well. More interestingly, in analogy with cavity induced selfordering we expect a corresponding phase transition also in the zero temperature limit, where the threshold is determined by the recoil energy and effective particle interaction strength replacing thermal energy. This should have distinctly different properties as compared to a 1D prescribed optical lattice. Obviously, even without the presence of the fiber, the interference of the scattered field by two distant atoms can induced long range forces and ordering. As part of the scattered field is lost, this leads to a higher threshold, but to some extend the particles itself can guide the scattered light. In the limit of sufficient initial particle density the trapped atoms themselves could form a fiber like guide for the scattered light enhancing long range interactions. One could speculate that the trapping field for the 1D confinement could be unnecessary and be provided by the light guided by the atoms. The predicted ordering phenomena could also be related to recent observations of collective scattering in dense atomic vapors\\cite{greenberg2011bunching,schmittberger2012free} where light induced selfordering mechanisms play an important role.\n\\par\nWe thank A. Rauschenbeutel, J. Kimble and I.Cirac for stimulating discussions and acknowledge support by the Austrian Science Fund FWF through the projects SFB FoQuS P13.\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nOff-shell currents are ordinarily used in efficient recursive evaluation of scattering amplitudes. These are computed from \n off-shell currents by setting \nthe external legs on-shell. Off-shell currents were famously employed by Berends and Giele \\cite{Berends:1987me} to recursively calculate tree-level gluon scattering and they are routinely used in matrix element generators \\cite{Gleisberg:2008fv, Cafarella:2007pc, Lifson:2022mxf}. Despite their success in practical calculations many of the appealing structures of on-shell scattering amplitudes are either not manifest \nor difficult to expose \\cite{Britto:2012qi, Mastrolia:2015maa, Jurado:2017xut} in off-shell currents. Similarly, in the classical limit of scattering amplitudes\\footnote{The classical of amplitudes has been received considerable attention in the last few years. The subject has been reviewed extensively in Refs.\\cite{Kosower:2022yvp,Buonanno:2022pgc}.} some of the on-shell features present before taking classical limit are not inherited. Off-shell currents in the classical limit have appeared recently in Refs.\\cite{Goldberger:2016iau,Goldberger:2017frp,Goldberger:2017ogt,Shen:2018ebu,Ben-Shahar:2021zww,Cristofoli:2020hnk}. \n\nIn this paper we are concerned with off-shell \ncurrents in the classical limit from the perspective of Kosower-Maybee-O'Connell (KMOC) \\cite{Kosower:2018adc, Maybee:2019jus, delaCruz:2020bbn, Cristofoli:2021vyo, Aoude:2021oqj}, \nthe Worldline Quantum Field Theory\n\\cite{Mogull:2020sak,Jakobsen:2021smu,Jakobsen:2021lvp, Jakobsen:2021zvh,Shi:2021qsb, Bastianelli:2021nbs,Wang:2022ntx} and their connection. \nThroughout, we will focus on the study of the classical limit of the tree-level off-shell current\n\\begin{align}\n\\mathcal{A}_n\n(p_1, p_2, k_1, \\dots, k_n)\\Big|_{k_i^2\\ne 0,\\, p_i^2=m^2} \\, ,\n\\end{align} \nwhich, assuming that $p_1$, $p_2$ are the momenta of two massive particles and $k_1, \\dots, k_n$ are the momenta of massless gauge bosons, can be fused into a $n+2$ scattering amplitude. Unlike typical currents in Berends-Giele recursions here more than one leg is off-shell. Similar objects have appeared previously in QCD, see e.g. Ref.\\cite{Schwinn:2005pi}. The \n$n=2$ current with spinning massive particles is relevant in the context of ongoing discussions of higher spin gravitational Compton amplitudes in the classical limit\n\\cite{Chiodaroli:2021eug,Johansson:2019dnu, Bautista:2021wfy, Saketh:2022wap, Bern:2022kto, Kosmopoulos:2021zoq,Jakobsen:2022zsx,Cangemi:2022abk,Bautista:2022wjf,Bern:2020buy, Cristofoli:2021jas,Aoude:2022thd}. \n\n\nThe remaining of the paper is organized as follows. In Section \\ref{off-shell-classical} we start with a general definition of a current and show how its classical limit is obtained in WQFT. In Section \\ref{computational-part} we calculate currents in QED and gravity.\nIn Section \\ref{currents-and-HTLs} we apply our methods to compute Hard Thermal Loops, which can be understood as classical objects in the high temperature regime. Our conclusions are presented in Section \\ref{conclusions}.\n\n\n\\section{Off-shell currents in the classical limit}\n\\label{off-shell-classical}\nLet us consider\na theory which models massive scalar particles coupled with massless gauge bosons. Let $p=(p_1, p_2)$ and $k=(k_1, \\dots, k_n)$ denote tuples of outgoing momenta of massive and massless external legs, respectively. We define the $n+2$ off-shell current by\n\\begin{align}\n\\mathcal{A}_n^{I_1, \\dots, I_n}\n(p, k):= \\hat{\\delta}^4(p_1+p_2+\\sum_{i=1}^n k_i)\nA_n^{I_1, \\dots, I_n}\n(p,k),\n\\label{general-def}\n\\end{align}\nwhere, without loss of generality \nthe massive scalars obey $p_i^2=m^2$, while for the massless particles $k_i^2\\ne 0$. We have introduced the notation $\\hat{\\delta}^n (x):=(2\\pi)^n (x)$ and for later use we define $\\hat{\\mathrm{d}}^n x:= (2\\pi)^n \\mathrm{d}^n x$. \n The upper indices denote collectively color and\/or Lorentz indices associated with the massless particles. \n\\begin{figure}\n\t\\centering\n\t\\begin{tikzpicture}[thick, transform shape]\n\t\\node[circle, fill=lightgray, draw] (c) at (0,0){$\\mathcal A$};\n\t\\draw[draw, snake it] (c.north west)--(-1.,1)node[left]\n\t{$\\qquad k_1$};\n\t\\draw[draw, snake it] (c.north east)--(1.,1)node[right]\n\t{$k_n$};\n\t\\draw[draw](1.5,-0.30)-- (c.south east)node[right, currarrow,\n\tpos=0.5, \n\txscale=1,\n\tsloped,\n\tscale=1]\n\t{};\n\t\\draw[draw] (-1.5,-0.30)--(c.south west)node[left, currarrow,\n\tpos=0.5, \n\txscale=-1,\n\tscale=1]{};\n\t\\node(A) at (0,1){$\\dots$};\n\t\t\\node(B) at (-1.5,0){$p_1$};\n\t \\node(C) at (1.5,0){$p_2$};\n\t\\end{tikzpicture}\n\t\\caption{Off-shell current. The blob represents a sum over tree-level Feynman diagrams where diagrams. Wavy lines represent off-shell outgoing massless gauge bosons.}\n\t\\label{current-off-shell}\n\\end{figure}\nFor our purposes the evaluation\nof this current will be through Feynman diagrams (see Fig.\\ref{current-off-shell}). The \ncurrent can be fused into an on-shell amplitude by\ndressing it with appropriate polarization vectors, imposing physical constraints such as transverse polarization, and setting all external legs on-shell,\nnamely\n\\begin{align}\n\\ii \\mathcal T \\sim \\mathcal \\xi_{I_1} \\cdots \\xi_{I_n}\\mathcal{A}_n^{I_1, \\dots, I_n}\n(p_1, p_2, k_1, \\dots, k_n)\\Big|_{k_i^2=0,p_i^2=m^2} \\,.\n\\end{align}\n\\subsection{Classical limit \\`a la KMOC}\nSuppose that we are interested in the classical limit of Eq.\\eqref{general-def} understood as the limit $\\hbar \\to 0$ or more precisely as a Laurent expansion in powers of some dimensionless parameter $\\xi$. \n In the spirit of KMOC one\n would restore $\\hbar$ into the current and perform dimensional analysis. One should also distinguish the momenta of massive scalar particles and massless gauge bosons. The latter being described by wavenumbers through the rescaling $k \\to \\hbar k$, while the former obeys\n $p_i^\\mu=m_i u_i^\\mu$, where $u_i^2=1$. To achieve definite classical momenta the initial states must be dressed with appropriate coherent\\footnote{ Here we think of coherent states in the sense of Perelemov \\cite{perelomov:1972}. We \n \trefer the interested reader to \\cite{Perelomov:1986tf} for details of the Perelomov formalism.} wavefunctions, giving us the notion of sharply peaked position and momenta. The most general coherent relativistic wavefunctions associated with the restricted Poincare group have the form \\cite{Kaiser:1977ys}\n %\n \\begin{align}\n f_z (p):=\\braket{e_z|f}=\n \\int \\mathrm{d} \\Phi(p) e^{-\\ii z\\cdot p} f(p) \\,,\n \\end{align} \n where $\\mathrm{d} \\Phi(p):= (2 \\pi)^{d-1}\\mathrm{d}^ d p \\delta (p^2-m^2)\\Theta(p_0)$ and \n $\\braket {e_z|p}:= \\mathcal C_z e^{-\\ii p \\cdot z}$. Here $z=x-\\ii y$ is a complex vector, which in general is time-dependent. \n The classical phase space is obtained by setting $t=0$ \\cite{Kowalski:2018xsw}. The normalization of states can be derived from\n\\begin{align}\n\\braket{e_z|e_w}= \\int \\mathrm{d} \\Phi (p) e^{-\\ii p\\cdot (z-\\bar w)}=\n \\mathcal C_ z \n\\mathcal C^{*}_w\n \\left(\\frac{m}{2\\pi \\eta}\\right)^{d\/2-1} \nK_{d\/2-1}(\\eta m), \n\\end{align}\nwhere $\\eta=\\sqrt{-(z-\\bar w)^2}$ and $K_\\nu (x)$ is the modified Bessel function.\n For $z=w$, $\\eta=2 |y|$, one obtains\n\\begin{align}\n\\mathcal C_z= \\left[{\\left(\\frac{4 \\pi |y|}{m} \\right)^{d\/2-1}\\frac{1}{K_{d\/2-1}(2m |y|)}}\\right]^{1\/2}.\n\\end{align}\n Wavefunctions employed by KMOC correspond to the case where one chooses the complex vector to be\n\\begin{align}\nz_i^\\mu=-b_i^\\mu+\\ii \\frac{ u_i^\\mu}{m \\xi}, \n\\end{align}\nwhere $\\xi$ is a dimensionless parameter, which can be thought as the square of the ratio of the Compton wavelength to the intrinsic spread of the wavepacket. Here $u_i$ is the classical four velocity of the particle of mass $m_i$.\nTherefore, it is natural to consider the current\\footnote{We will keep\n\tthe indices explicit for calculations but otherwise suppress them to avoid cluttered expressions.} weighted with coherent wavefunctions as \n\\begin{align}\n\\label{current1}\n&\\mathcal{C}_n (p, k):=\\int \\mathrm{d} \\Phi (p_1, p_2) \\phi_{z_1}(p_1) \\phi_{z_2} (p_2) \\mathcal{A}_n\n(p_1, p_2, k_1, \\dots, k_n) \\, ,\n\\end{align} \nwhere $ \\mathrm{d} \\Phi (p_1, p_2):=\\mathrm{d} \\Phi(p_1) \\mathrm{d} \\Phi(p_2)$. Now the classical limit of the current can\n be computed from \n a Laurent expansion of the formal expression\n\\begin{align}\n\\label{current}\n&\\mathcal{C}_n (p, k)=\\int \\mathrm{d} \\Phi (p_1, p_2) \\phi (p_1) \\phi (p_2) e^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\mathcal{A}_n\n(p_1, p_2, \\hbar k_1, \\dots, \\hbar k_n) \\, ,\n\\end{align} \nwhere we have used momentum conservation, $k\\to\\hbar k$ and $b \\to b\/\\hbar$. The rescaling has an important effect into the structure of \\eqref{current}. Indeed writing explicitly the momentum conservation Dirac-delta we have\n\\begin{align}\n&\\mathcal{C}_n (p, k)= \n\\int \\mathrm{d} \\Phi (p_1, p_2) \\phi (p_1) \\phi(p_2) e^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\hat{\\delta}^4(p_1+p_2+\\hbar \\sum_{i=1}^nk_i ) { A}_{n} (p,\\hbar k)\\\\ \n& = \n\\int \\mathrm{d} \\Phi(p_1) \\ \\phi (p_1)\n\\phi(-p_1 - \\hbar \\sum_{i=1}^n k_i )\n\\hat{\\delta}( \\hbar^{2} \\sum_{i,j=1}^{n} k_{i}\\cdot k_{j} +2 \\hbar p_1 \\cdot \\sum_{i=1}^n k_i ) e^{\\ii b \\cdot \\sum_{i=1}^n k_i} {A}_{n}(p,\\hbar k), \\nonumber\n\\end{align}\nwhere we have used the momentum conservation Dirac-delta to perform the phase-space integral over $p_2$.\nMaking the identifications $q \\to \\sum_{i=1}^n k_i$ the remaining integral has the form\n\\begin{align}\n\\int \\mathrm{d} \\Phi(p) \\phi (p)\n\\phi (-p- \\hbar q) \\hat{\\delta}(2\\hbar p \\cdot q+\\hbar^2 q^2) f(p, q),\n\\end{align} \nwhich is sharply peaked around $p^\\mu=m u^\\mu$, where $u^\\mu$ is the classical velocity of the particle with mass $m$. The analysis of the above integral by KMOC (see Appendix B of Ref.\\cite{Kosower:2018adc}) does not depend on the on-shell properties of \n$q$, which plays the role of the momentum mismatch in KMOC, so we can apply it here as well. The current then simplifies to\n\\begin{align}\n\\mathcal{C}_n (p, k)= \\frac{1}{2}\\hat{\\delta}\\left( p \\cdot \\sum_{i=1}^n k_i\\right)\ne^{\\ii b \\cdot \\sum_{i=1}^n k_i}\n\\bar { A}_{n} (p,k) \\,, \n\\label{final-current-KMOC}\n\\end{align}\nwhere $\\bar A(p,k)$ denotes the non-vanishing term in the Laurent expansion, where by abuse of notation we have set $p_{1}=p$. From a practical point of view we are done. We can already perform calculations following the KMOC algorithm and reach Eq.\\eqref{final-current-KMOC} for the theory under study.\n\nA few comments are in order. Strictly speaking Eq.\\eqref{final-current-KMOC} should be understood as the average of the RHS\nover wavefunction of $p$, which in KMOC is denoted by a double bracket. \n The net effect of weighting over coherent wavefunctions is producing an overall factor depending on the external soft momenta, which is analogous to the momentum mismatch in classical observables. \n The attentive reader might ask about the presence of singular terms produced by the series expansion.\nThe current is not an observable so one might expect such terms. However, the current is an off-shell tree-level object so we may safely ignore Feynman's $\\ii \\epsilon$ prescription in calculations, thus leading to cancellation of those singular terms. In QED we have checked this up to 7-points. In the worldline formulation of the\ncurrent the absence of those singular terms will become clear as we discuss now.\n \n\\subsection{Relation to WQFT }\\label{sec-proof-WQFT}\n\nWe wish to relate the classical current \\eqref{final-current-KMOC} to a worldline path integral. Although we will consider scalar QED as an illustration, the following discussion is also applicable to other theories where the WQFT is known. The field theory \naction for scalar electrodynamics \n\\begin{align}\nS_{\\text{sQED}}[\\varphi,\\varphi^{\\dagger},A] =\\int \\mathrm{d}^4 x\\left[(D^\\mu\\varphi)^\\dagger D_\\mu \\varphi -m^2 \\varphi^\\dagger \\varphi\\right] \\, \n\\end{align}\nwith the gauge covariant derivative $D_\\mu=\\partial_\\mu-\\ii e A_\\mu$. The gauge-fixed Maxwell action is\n\\begin{align}\nS_{\\text{gf}}[A]=- \\int \\mathrm{d}^4 x \\left[\\frac 1 4 F_{\\mu\\nu}F^{\\mu\\nu}+\\frac 1 {2\\xi } (\\partial^\\mu A_\\mu)^2\\right],\n\\end{align}\nwhere we will use Feynman gauge ($\\xi=1$) in in calculations. We know from standard QFT that the current \\eqref{general-def} can be written off-shell as a Fourier transform of a time ordered correlation function of the quantum fields describing the external states after LSZ reduction, which can be represented as a path integral. Moreover, we are interested in the classical limit so it will be convenient to perform the LSZ reduction in two steps, first on the photon legs and then on the scalar ones. \n In momentum space the amputation of the photon external legs leads to\n\\begin{align} \\label{P-off}\n{\\cal P}_n^{\\text{sQED}}(p,k )\n=\\frac{1}{\\cal N}\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]} \\int& {\\cal D}\\varphi {\\cal D} \\varphi^{\\dagger } e^{\\ii S_{\\text{sQED}}[\\varphi,\\varphi^{\\dagger},A]}\\\\\n& \\ii k_1^2 \\cdots \\ii k_n^2\\ \\varphi(p_{1})\\varphi^{\\dagger}(p_{2}) A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\n\\Big|_{k^2\\ne 0} \\nonumber\n \\,.\n\\end{align}\n\nBefore LSZ let us consider the classical limit of Eq.\\eqref{P-off} following Ref.~\\cite{Mogull:2020sak}. In the classical approximation we can neglect loops mediated by scalars and replace the path integral over scalar fields by the photon-dressed scalar propagator, which we briefly review now\\footnote{Dressed propagators have been developed\n\tin a worldline representation for a variety of models, see e.g. \\cite{Ahmadiniaz:2015kfq,\n\t\tAhmadiniaz:2015xoa, Edwards:2017bte, Ahmadiniaz:2017rrk, Ahmadiniaz:2020wlm, Corradini:2020prz,\n\t\tAhmadiniaz:2021gsd}. A standard reference for worldline methods is Ref.\\cite{Schubert:2001he}.}.\n\tIn position space the photon-dressed scalar propagator reads\n\\begin{equation}\n\\mathcal G(x_1,x_2)[A] = \\frac{1}{{\\bar{ N}}}\\int {\\cal D} \\varphi {\\cal D}\\varphi^{\\dagger} e^{\\ii S_{\\text{sQED}}[\\varphi,\\varphi^{\\dagger},A]}\n\\varphi(x_{1})\\varphi^{\\dagger}(x_{2})\\,.\n\\end{equation}\nThe dressed propagator admits a worldline path integral representation of the form\n\\begin{align}\n\\mathcal G(x_1,x_2)[A] &= \\langle x_{2}|\\frac{1}{D^{2} +m^{2} - \\ii\\epsilon }| x_{1}\\rangle \\\\\\nonumber\n&= \\ii \\int_{0}^{\\infty}\\mathrm{d} T \\,\\langle x_{2}| e^{-\\ii T (D^{2} +m^{2} + \\ii\\epsilon )}| x_{1}\\rangle\n= \\int_{0}^{\\infty } \\mathrm{d} T \\int_{x(0)=x_{1}}^{x(1)=x_{2}} {\\cal D} x \\, e^{-\\ii \\int_{0}^{1}d\\tau \\left( \\frac{1}{2T}\\dot{x}^{2} +e \\dot{x}_{\\mu}A^{\\mu}\\right)}\\,,\n\\end{align} \nwhere $T$ is the so-called Schwinger proper time while the $\\ii\\epsilon$ prescription is understood in the path integral. The dressed propagator in momentum space can be written as \\cite{Daikouji:1995dz}\n\\begin{align}\n\\mathcal G(p,k)[A]=\n\\hat{\\delta}^4 \\left(p_1+p_2+\\sum_{i=1}^n k_i\\right) G (p,k)[A]\\,,\n\\label{dressed-gen}\n\\end{align} \nwhose explicit form is not required for our purposes. Hence the path integral with no matter loops is\n\\begin{align} \\label{P-off2}\n{\\cal P}_n^{\\text{sQED}}(p,k )\n=&\\frac{1}{\\tilde {\\cal N}}\n\\hat{\\delta}^4 \\left(p_1+p_2+\\sum_{i=1}^n k_i\\right)\\\\\n&\n\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]} G (p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\\Big|_{k^2\\ne 0}^{\\text{trees}} \\nonumber\n\\,,\n\\end{align}\nwhere we have absorbed factors related to our definition of dressed propagator into $\\tilde {\\mathcal N}$. This is not yet the classical limit of \nthe current we are looking for because we have not performed the LSZ reduction on the external scalar legs, namely \t\\cite{Fabbrichesi:1993kz}\n \\begin{align}\n G^{\\text{c}}(p)[A]= \\lim\\limits_{p_1^2, p_2^2 \\to m^2} \\ii(p_1^2-m^2) \\ \\ii (p_2^2-m^2) \\int \\mathrm{d}^4 x \\mathrm{d}^4y \\\n e^{\\ii p_1\\cdot x+\\ii p_2\\cdot y}G(x, y)[A] \\;.\n \\label{Gconnected}\n \\end{align}\n At the level of the worldline integral we can also achieve the LSZ reduction by\n Fradkin's prescription of exchanging the limit of integration in the worldline action to $(-\\infty, +\\infty)$ \n as a consequence of performing the Schwinger proper time integration after amputating the external scalar propagators. \n The latter step fixes the boundary conditions needed to perform the perturbative expansion of the amputated dressed propagator. The equivalence between both procedures is encoded in the relation\n \\begin{align} \\frac{\\Sigma(b,p;A)}{\\Sigma_{0}}=\n e^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\hat{\\delta}\\left( p\\cdot \\sum_{i=1}^{n}k_{i}\t\\right)G^c(p;A) \\,, \n \\label{factors-MPS}\n \\end{align}\n which was \n explicitly demonstrated for the graviton-dressed scalar propagator in Ref.\\cite{Mogull:2020sak}. Here $\\Sigma_{0}$ is some overall factor that we can absorb into the normalization of the correlation function. Notice that both sides depend only on $p_1=p$ on the support of the Dirac-delta in Eq.\\eqref{dressed-gen}. The left hand side of Eq.\\eqref{factors-MPS} is given by\n \\begin{align}\n \\Sigma(b,p;A) = \\int {\\cal D} x \\exp \\left[ \n -\\ii \\int_{-\\infty}^{\\infty}d\\tau \\left( \\frac{1}{2}\\dot{x}^{2} + e \\dot{x}_{\\mu}A^{\\mu}(x(\\tau))\t\t\t\t\\right)\\right], \n \\label{sigma-path}\n \\end{align} \nwhere $b$ and $p$ arise\n from the background expansion $x^{\\mu}(\\tau)= b^{\\mu} + p^{\\mu}\\tau +q^{\\mu}(\\tau)$. We will see in later Sections how to evaluate \\eqref{sigma-path} from Worldline Feynman Rules (WFRs). \n \n Going back to Eq.\\eqref{Gconnected} and restricting the calculation to tree-level, the current can be written as\n \\begin{align} \\label{P-off3}\n {\\cal A}_n^{\\text{sQED}}(p,k )\n =&\\frac{1}{\\tilde {\\cal N}}\n \\hat{\\delta}^4 \\left(p_1+p_2+\\sum_i k_i\\right)\\\\\n &\n \\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]} G^c(p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\n \\Big|_{k^2\\ne 0, p_i^2 =m^2}^{\\text{trees}} \\nonumber\n \\,,\n \\end{align} \n which brings us closer to the WQFT representation of $\\mathcal C^{\\text{sQED}}_n (p,k)$. Recalling Eq.\\eqref{general-def} and imposing momentum conservation we have\n\\begin{align}\n\\bar A(p,k)=\\frac{1}{\\tilde {\\cal N}}\n\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]} G^c(p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\\Big|_{p_2=-p_1-\\sum_i k_i} \\, ,\n\\end{align}\nwhich is a purely tree-level object in the classical approximation which we can identify \nwith $\\bar A(p,k)$ in\nEq.\\eqref{final-current-KMOC}. Thus, inserting this equation into Eq.\\eqref{final-current-KMOC} we have\n\\begin{align}\n\\mathcal{C}^{\\text{sQED}}_n (p, k)=&\n\\frac{1}{\\tilde {\\cal N}}\ne^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\hat{\\delta}\\left( p\\cdot \\sum_{i=1}^{n}k_{i}\t\\right)\\\\\n&\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]} G^c(p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\\Big|_{p_2=-p_1-\\sum_i k_i} \\, .\\nonumber\n\\end{align}\nFinally, applying the relation \\eqref{sigma-path} \nwe arrive at the WQFT representation of the off-shell current\n\\begin{align}\n\\mathcal{C}^{\\text{sQED}}_n (p, k)=\n\\frac{1}{\\cal Z} \n\\int {\\cal D} A \\, e^{\\ii S_{\\text{gf}}[A]} \\,\n\\int {\\cal D} x e^{\n\t-\\ii \\int_{-\\infty}^{\\infty}d\\tau \\left[ \\frac{1}{2}\\dot{x}^{2} + e \\dot{x}_{\\mu}A^{\\mu}(x(\\tau))\t\t\t\t\\right]} \\, \\ii k_1^2 A_{\\mu_{1}}(k_{1})\\cdots \\ii k_n^2A_{\\mu_{n}}(k_{n}) ,\n\\label{path-integral-exp}\n\\end{align}\nwhich generalizes WQFT for an arbitrary number of off-shell photons\\footnote{Another alternative to generate photon insertions is to consider the functional with a factor $\\exp(JA)$ and take derivatives over $J$.}. \nThe factor\n${\\cal Z}$ ensures that the current is normalized to one when there are no gauge fields in the path integral and defines the WQFT partition function\n\\begin{equation}\\label{zqed}\n{\\cal Z} = \\int {\\cal D} A \\, e^{\\ii S_{\\text{gf}}[A]} \\, \\int {\\cal D} x \\, e^{-\\ii \\int_{-\\infty}^{\\infty} \\mathrm{d}\\tau \\left(\t\\frac{1}{2}\\dot{x}^{2} + e \\dot{x}\\cdot A\t\\right) } \\, .\n\\end{equation}\nThe boundary conditions on the worldline variables depend on the model under consideration. Collecting all of the integration constants into $\\mathcal Z$ the classical off-shell current in WQFT can be succinctly written as\n\\ba \n\\mathcal{C}^{\\text{sQED}}_n (p, k)&= \\ii k_1^2 \\cdots \\ii k_n^2\\left \\langle A_{\\mu_{1}}(k_{1})\\hdots A_{\\mu_{n}}(k_{n})\\right\\rangle_{\\text{WQFT}}\\, .\n\\label{current-classical-worldlined-QED}\n\\ea \nCalculations based on Eq.\\eqref{path-integral-exp} will produce precisely the factors in Eq.\\eqref{final-current-KMOC} from which we can\nextract $\\bar A_n(p,k)$. In doing so one must keep in mind the overall factor of two in Eq.\\eqref{final-current-KMOC}. Moreover, the presence of the Dirac-delta is required to match currents in both approaches but can be dropped at the end of the computation. The worldline path integral \\eqref{current-classical-worldlined-QED} can be treated perturbatively deriving Feynman rules from the worldline action leading directly to $\\bar A_n(p,k)$ without any Laurent expansion. \n\nOther theories can be treated along the same lines. For gravity we will consider this in Section \\ref{computational-part}. \nGauge theories with color can be treated along the same lines with the usual caveats surrounding properties of color charges in the classical limit. On the amplitudes side one can follow\nRef.\\cite{delaCruz:2020bbn} while in the WQFT front one can apply the methods of Ref.\\cite{Shi:2021qsb}. \n \n\n\\section{Computing off-shell currents}\n\\label{computational-part}\nThe evaluation of Eq.\\eqref{current-classical-worldlined-QED} can be done by deriving worldline Feynman rules (WFRs), which take care of the perturbative evaluation of the Gaussian path integral. There are two types of Wick contractions: those related with the worldline path integral and Wick contractions among gauge fields.\nThe net effect of the contraction among fields will be to generate permutations. \n\nThe partition functions considered in the rest of the paper have the schematic form\n %\n \\begin{align}\n \\mathcal Z = \\int {\\mathcal D} B\\, e^{\\ii S[B]}\n \\int {\\mathcal D} x \\int \\mathcal D \\chi e^{\\ii S\\left[x, \\chi; B (x)\\right]}\\, ,\n \\end{align}\n where $B$ represents a background field and $\\chi$ are auxiliary worldline variables. The action $S[B]$ is the gauge fixed action of the background field and \n $S\\left[x, \\chi; B (x)\\right]$ is the worldline action in WQFT, from which we can derive WFRs. \n We rescale the worldline parameter to derive WFRs written in terms of the worldline momentum $p^{\\mu}$ instead of $u^\\mu$. The relevant wordline actions are\n \\begin{align}\n S_{\\text{BA}}[x,\\lambda,\\bar{\\lambda},\\gamma,\\bar{\\gamma}; \\Phi]=&- \\int_{-\\infty}^\\infty \\mathrm{d} \\sigma\\left(\\frac 1 2 \\dot x ^2+\\ii \\bar \\lambda_a \\dot \\lambda^a +\\ii \\bar \\gamma_\\alpha \\dot \\gamma^\\alpha - \\frac {y} 2 Q^a\\Phi^{a\\alpha} \\tilde Q^\\alpha \\right)\\, , \\label{BAction}\\\\\n S_{\\text{YM}}[x, \\lambda; A] =& -\\int_{-\\infty}^{\\infty}\\mathrm{d}\\sigma\\left(\n \\frac{1}{2}\\dot{x}^{2}+\\ii \\bar{\\lambda}_{a}\\dot{\\lambda}^{a}+g \\dot{x}^{\\mu}A_{\\mu}^{a}Q^{a} \\right)\\,, \\label{YMC}\\\\\n\tS_{\\text{GR}}[x, \\psi, \\bar \\psi;A]=& -\\int_{-\\infty}^{\\infty} \\mathrm{d}\\sigma \\left( \\frac{1}{2}g_{\\mu\\nu}\\dot{x}^{\\mu}\\dot{x}^{\\nu} + \\ii \\bar{\\psi}_{a} \\frac{D \\psi^{a}}{D\\tau} - \\frac{1}{8}R_{abcd}S^{ab}S^{cd}\t\t\t\t\\right)\\, ,\\label{GRSpin}\n\\end{align}\nfor Bi-Adjoints, Yang-Mills, and GR, respectively.\nThe covariant derivative is $\\frac{D\\psi^{a}}{D\\tau} = \\dot{\\psi}^{a} + \\dot{x}^{\\mu}\\omega_{\\mu a b}\\psi^{b}$ and $\\omega_{\\mu a b}= e_{a\\nu}(\\partial_{\\mu}e^{\\nu}_{b} + \\Gamma^{\\nu}_{\\mu\\lambda} e^{\\lambda}_{b} )$ is the spin connection. The translation between curved and flat indices is done by using the tetrad fields $e^{\\mu}_{a}$.\nThe spin tensor of the particle is defined by $S^{\\mu\\nu} = -2\\ii e^{\\mu}_{a} e^{\\nu}_{b}\\ \\bar{\\psi}^{[a}\\psi^{b]}$.\nThe derivation of WFRs is now well established and described at length\nin Refs. \\cite{Mogull:2020sak, Jakobsen:2021zvh, Shi:2021qsb, Bastianelli:2021nbs, Wang:2022ntx} so here we only consider briefly EM. Our treatment of color variables is equivalent to Ref.\\cite{Shi:2021qsb} but differs on the introduction of auxiliary color variables\\footnote{A pedagogical description of auxiliary variables is in Ref.\\cite{Bastianelli:2021rbt}.}.\n The interested reader can see the details of the derivation of WFRs with color in Appendices \\ref{bi-adjoint-WQFT} and \\ref{colors-appendix}.\n\n \\subsection{Gauge} \n %\n \\label{gauge-examples}\nWFRs are easy to derive for the colorless version of \n\\eqref{YMC}. \n Inserting the background expansion $x^\\mu (\\tau)= b^{\\mu} + p^{\\mu}\\tau + q^{\\mu}(\\tau)$ into the action\n\\begin{align}\nS_{\\text{sQED}}[x; A] =& -\\int_{-\\infty}^{\\infty}\\mathrm{d}\\sigma\\left(\n\\frac{1}{2}\\dot{x}^{2}+e \\dot{x}^{\\mu}A_{\\mu} \n\\right)\\,,\n\\end{align}\nwhere \n\\begin{align}\nq^{\\mu}(\\tau) = \\int_{-\\infty}^{\\infty}\\hat{\\mathrm{d}}\\omega \\, e^{\\ii \\omega \\tau}q^{\\mu}(-\\omega), \\ \\hskip.5cm A^\\mu (x) = \\int_{-\\infty}^{\\infty}\\hat{\\mathrm{d}}^4 k \\, e^{\\ii k \\cdot x }\nA(-k),\n\\end{align} \n the propagators of the worldline and the gauge field (inf Feynman gauge) are\n\\begin{align} \\label{wprop}\n\\raisebox{-1.7mm}{\n\t\\begin{tikzpicture}[thick]\n\t\\coordinate (A) at (-1,-0);\n\t\\coordinate (B) at (1,-0);\n\t\\filldraw (A) circle (2pt) node[left] {$q^\\mu$};\n\t\\filldraw (B) circle (2pt) node[right] {$q^\\nu$};\n\t\\draw[] (0,.5) node[]{$\\omega$};\n\t\\path[ very thick,draw] (-1,0)--(1,0); \t\\path[ thick,draw,->] (-0.5,0.3)--(0.5,0.3) node[pos=.5] {}; \n\t\\end{tikzpicture}\n}\n= - \\ii \\frac{\\eta^{\\mu\\nu}}{(\\omega+\\ii\\epsilon)^2},\n\\hskip.4cm \n\\raisebox{-1.7mm}{\n\t\\begin{tikzpicture}[thick]\n\t\\coordinate (A) at (-1,-0);\n\t\\coordinate (B) at (1,-0);\n\t\\filldraw (A) circle (2pt) node[left] {$A^\\mu$};\n\t\\filldraw (B) circle (2pt) node[right] {$A^\\nu$};\n\t\\draw[] (0,.6) node[]{$k$};\n\n\t\\path[ thick,draw,->] (-0.5,0.3)--(0.5,0.3) node[pos=.5] {}; \n\t\\path[snake it,draw] (-1,0)--(1,0); \n\t\\end{tikzpicture}\n}\n= - \\ii \\frac{\\eta^{\\mu\\nu}}{(k^{2}+\\ii\\epsilon)} \\, . \n\\end{align}\n In addition, the interaction vertices propagating $n$-quantum fluctuations of the worldline variables can be compactly written as (see also Ref.\\cite{Wang:2022ntx})\n\\begin{equation}\n\\raisebox{-10mm}{\\vnQED } = e\\,\\ii^{n-1} e^{\\ii b\\cdot k}\\hat{\\delta}\\left(k\\cdot p +\\sum_{i=1}^{n} \\omega_{i}\\right) \\left(\np^{\\mu}\\prod_{i=1}^{n}k^{\\alpha_{i}} + \\sum_{i=1}^{n}\\omega_{i}\\eta^{\\mu \\alpha_{i}}\\prod_{j\\neq i}^{n}k^{\\alpha_{j}}\n\\right)\\,, \n\\end{equation}\nwhere the solid lines are outgoing. Contractions of gauge fields generate topologies of worldline diagrams, in which external propagators must be amputated in the sense of LSZ. For instance at lower orders it is easy to see that the Wick contractions among gauge fields lead us to evaluate the diagrams shown in Fig.\\ref{main-topologies-off-shell}.\n\n\n\\subsubsection*{Examples}\n\n \\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\\centering\t\\lotop{$A_1$}{$A_2$}{black}\n\t\t\\caption{}\n\t\t\\label{lo-gen}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\\centering\n\t\t\\nlotop{$A_1$}{$A_2$}{$A_3$}{black}{black}\n\t\t\\caption{}\n\t\t\\label{nlo-gen}\n\t\\end{subfigure}\n\t\\caption{Worldline diagrams required for the calculation of 2-point (a) and 3-point (b) currents. The dashed lines are drawn only for illustration and represent the worldline. Solid lines represent fluctuations of matter lines and later also auxiliary variables. Wavy lines represent off-shell massless particles and $A$ its associated field. }\n\t\\label{main-topologies-off-shell}\n\\end{figure}\nLet us consider the $n=2$ off-shell current for scalar electrodynamics.\nTwo equivalent diagrams with symmetry factor $\\frac 12$ are generated by the wordline path integration. Hence it is enough to consider the one in Fig.\\ref{lo-gen} whose symmetry factor is unity.\nAn easy calculation gives\n\\ba \n\\mathcal{C}_{\\text{sQED}}^{\\mu\\nu} (p, k) &=e^2 \\, e^{\\ii b\\cdot (k_1+k_2)} \n \\hat{\\delta}\\left(p\\cdot (k_{1}+k_{2})\\right) \\bar A_{\\text{sQED}}^{\\mu\\nu}(k_{1},k_{2})\\\\\n&=e^2 \\, e^{\\ii b\\cdot (k_1+k_2)}\\hat{\\delta}\\left(p\\cdot (k_{1}+k_{2}) \\right) \\ii \\left(\n \\eta ^{\\mu \\nu }+\\frac{k_2^{\\mu } p^{\\nu }}{p\\cdot k_{1} }-\\frac{ k_1^{\\nu } p^{\\mu }}{p\\cdot k_{1} }-\\frac{ k_1\\cdot k_2\n p^{\\mu } p^{\\nu }}{(p\\cdot k_{1})^2}\n \\right),\n \\label{2ptsQED}\n\\ea \nwhich satisfies the Ward identity $k_{i,\\mu} \\mathcal{C}_{\\text{sQED}}^{\\mu\\nu} (p, k) = 0$ and matches what one calculates using Eq.\\eqref{final-current-KMOC}. A nontrivial example is the $n=5$ case. Some topologies involved are shown in Fig.\\ref{topologies-five-photons}. Its Feynman diagrammatic calculation requires 450 diagrams but the worldline calculation effectively requires only 12 diagrams summed over the permutations of the external photons with no need to perform any Laurent expansion in $\\hbar$, so its calculation is simpler in WQFT. The result obtained with WQFT is lengthy\\footnote{The full expression for $\\bar A_{\\text{sQED}}^{\\mu_1\\dots\\mu_5}$ is attached to this submission in FeynCalc notation.}. Schematically it can written as \n %\n \\begin{align}\n \\mathcal{C}_{\\text{sQED}}^{\\mu_1 \\dots \\mu_5}=e^5\\, e^{\\ii b\\cdot (k_1+\\dots +k_5)} \\hat{\\delta} \\left(p\\cdot \\sum _{i=1}^5 k_i\\right) \\bar A_{\\text{sQED}}^{\\mu_1\\dots\\mu_5}=e^5\\, e^{\\ii b\\cdot (k_1+\\dots +k_5)} \\hat{\\delta} \\left(p\\cdot \\sum _{i=1}^5 k_i\\right) \\sum_{i=1}^{2451} a_i \\mathsf{T}_i^{\\mu_1 \\dots \\mu_5},\n \\label{eq-5-pt}\n \\end{align}\n where the sum runs over independent tensor structures. We have compared this result against the Feynman diagram calculation and found agreement. \n \n \n \\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[t]{0.4\\textwidth}\n\t\t\\centering\t\\lotopfivePHone{$A^{\\mu_1}$}{$A^{\\mu_2}$}{$A^{\\mu_3}$}{$A^{\\mu_4}$}{$A^{\\mu_5}$}\n\t\t\\caption{$S=4!$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\\centering\t\\lotopfivePHtwo{$A^{\\mu_1}$}{$A^{\\mu_2}$}{$A^{\\mu_3}$}{$A^{\\mu_4}$}{$A^{\\mu_5}$}\n\t\t\\caption{$S= 3!$}\n\t\\end{subfigure}\\\\\n\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\\centering\t\\lotopfivePHthree{$A^{\\mu_1}$}{$A^{\\mu_2}$}{$A^{\\mu_3}$}{$A^{\\mu_4}$}{$A^{\\mu_5}$}\n\t\t\\caption{$S=2 \\times 3!$}\n\t\\end{subfigure}\n\t\\caption{Examples of worldline topologies required for the calculation of 7-point sQED current and their symmetry factors $S$. }\n\t\\label{topologies-five-photons}\n\\end{figure}\n\n\n\n \n\\subsection{Gravity}\n\\label{gr-off-shell}\n The worldline QFT allows us to introduce classical spin effects at almost no cost so we will do so. Let us briefly\nsummarize the WQFT approach of Ref.\\cite{Jakobsen:2021zvh}, which employs\n the so called ${\\cal N} = 2$ model. This model captures quadratic in spin effects in classical scattering through the introduction of complex Grassman variables $\\psi,\\bar{\\psi}$ allowing to gauge two super-symmetries on the worldline. The relevant actions to construct the partition function are \n %\n \\begin{equation}\n S_{\\textrm{EH}}[g] = -\\frac {2}{\\kappa^{2}}\\int \\mathrm{d}^{4}x\\, \\sqrt{-g}\\, R, \\quad S_{\\text{gf}} = \\int \\mathrm{d}^{4}x \\left(\t\\partial^{\\nu}h_{\\nu\\mu}-\\frac 1 2 \\partial_{\\mu} h^{\\nu}_{\\nu}\\right)^{2},\n \\end{equation}\nwhere the space time metric is perturbatively expanded as $g_{\\mu\\nu}= \\eta_{\\mu\\nu}+\\kappa h_{\\mu\\nu}$. The gauge-fixing term\n imposes a weighted version of the de Donder gauge $\\partial^{\\nu}h_{\\nu\\mu} = \\frac 12 \\partial_{\\mu} h^{\\nu}_{\\nu}$, which leads to the graviton\n propagator\n \\begin{align}\n \\label{eq:propq}\n \\raisebox{-2.8mm}{\n \t\\begin{tikzpicture}[thick]\n \t\\coordinate (A) at (-1,-0);\n \t\\coordinate (B) at (1,-0);\n \t\\filldraw (A) circle (2pt) node[left] {$h_{\\mu\\nu}$};\n \t\\filldraw (B) circle (2pt) node[right] {$h_{\\rho\\sigma}$};\n \t\\draw[] (0,.6) node[]{$k$};\n \n \t\\path[ thick,draw,->] (-0.5,0.3)--(0.5,0.3) node[pos=.5] {}; \n \t\\path[double,snake it,draw] (-1,0)--(1,0); \n \t\\end{tikzpicture}\n }\n = \\frac{\\ii}{2k^{2}} \\left(\t \\eta_{\\mu \\rho}\\eta_{\\nu \\sigma} + \\eta_{\\mu \\sigma}\\eta_{\\rho\\nu} - \\eta_{\\mu\\nu}\\eta_{\\rho\\sigma} \t\\right) \\;.\n \\end{align}\n %\nThe partition function can be written as follows\n\\begin{equation} \\label{zgr}\n{\\cal Z} = \\int {\\cal D} h_{\\mu\\nu}\\, e^{\\ii (S_{\\textrm{EH}} +S_{\\textrm{gf}} ) } \\int {\\cal D} x {\\cal D} \\psi {\\cal D} \\bar{\\psi} \\, e^{\\ii S_{\\text{GR}}},\\end{equation}\nwith the worldline action given in \\eqref{GRSpin}. The worldline path integral measure includes Lee-Yang ghost fields\n\\begin{equation}\n{\\cal D} x = \\int Dx DaDbDc\\, \\exp\\left(\t-\\frac{\\ii}{2}\\int_{-\\infty}^{\\infty}\\mathrm{d}\\sigma\\, g_{\\mu\\nu}\\left(a^{\\mu}a^{\\nu} + b^{\\mu}c^{\\nu}\t\t\\right)\t\t\\right)\n\\end{equation}\nwith $a$ bosonic and $b,c $ fermionic, which make the measure translational invariant. For details on the model we refer the reader to \\cite{Jakobsen:2021zvh}.\nTo derive WFRs in such a case one also needs to background expand $\\psi^{a}(\\tau) = \\theta^{a}+ \\Psi^{a}(\\tau)$ where complex conjugation leads to the expansion for the barred partner. This yields the classical value of the spin tensor ${\\cal S}^{ab} = -2\\ii \\bar{\\theta}^{[a}\\theta^{b]}$, which is related to the Pauli-Lubanski spin vector $s^{a} = -\\frac{1}{2}\\epsilon^{abcd}p_{b}{\\cal S}_{cd}$. The required WFRs to perform calculations are given in the Appendix \\ref{WFR-gr}. Generalizing the results in Sec.\\ref{sec-proof-WQFT}, we write the off-shell current in gravity as\n\\begin{equation}\n{\\cal C}_n^{\\text{GR}}(p,k) = (- \\ii)^{n} k_{1}^{2} k_{2}^{2} \\cdots k_{n}^{2}\\braket{ h_{\\mu_{1} \\nu_{1}}(k_{1}) h_{\\mu_{2} \\nu_{2}}(k_{2}) \\cdots h_{\\mu_{n} \\nu_{n}}(k_{n})\n}_{\\text{WQFT}}\n\\label{current-scalar-GR}\n\\end{equation} \nunderstood as an expectation value over the partition function \\eqref{zgr}. \n\nAs an example we consider the classical on-shell Compton amplitude\\footnote{In Ref.\\cite{Saketh:2022wap} a direct classical calculation of the Compton amplitude has been done using the worldline effective theory including cubic in spin corrections.}, which describes the scattering of linearized gravitational waves off a black hole. \n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\t\\centering \\lotopGR{$h_{\\mu\\nu}$}{$h_{\\alpha\\beta}$}{black}\n\t\t\\caption{}\n\t\t\\label{tu} \t\t\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\t\\centering \\lotopGRspin{$h_{\\mu\\nu}$}{$h_{\\rho\\sigma}$}{black}\n\t\t\t\\caption{}\n\t\t\\label{tu1} \n\t\t\\end{subfigure}\\\\[1em]\n\t\t\\begin{subfigure}[t]{.3\\textwidth}\n\t\t \\centering \\hhh\n\t\t \\caption{} \n\t\t \\label{s}\n\t\t \\end{subfigure}\n\t\t \\begin{subfigure}[t]{.3\\textwidth} \n\t\t\\centering \\hhvertex \n\t\t\\caption{}\n\t\t\\label{seagull} \n\t\\end{subfigure}\n\t\\caption{Worldline diagrams contributing to the on-shell Compton amplitude. \\ref{tu},\\ref{tu1}, with crossed topologies are associated with fluctuations of kinematic and spin variables, respectively.\n\t }\n\t\\label{compton-amplitude}\n\\end{figure}\nThe on-shell Compton amplitude is given by\n\\begin{equation}\\label{sGR-Compton}\n{\\cal M}^{h_{1}h_{2}}(p,k_{1},k_{2}) = \n{\\bar M}_{\\text{GR}}^{\\mu\\nu\\alpha\\beta} (p, k_{1},k_{2}) \\epsilon^{h_{1}}_{\\mu\\nu}(k_{1})\\epsilon^{h_{2}}_{\\alpha\\beta}(k_{2})\\Big|_{k_i^2=0}\\,,\n\\end{equation}\nwhere we can extract the amplitude from $\\mathcal{C}_{\\text{sGR}}^{\\mu\\nu\\alpha\\beta}=\\frac{1}{2}\\kappa^2 e^{\\ii b\\cdot (k_1+k_2)} \\hat{\\delta}(p\\cdot (k_1+k_2)) \\bar M_\\text{GR}^{\\mu\\nu\\alpha\\beta}$, which receives contributions from the diagrams shown in Fig.\\ref{compton-amplitude}. \nIn the spinless case, we can apply and off-shell version of the the Kawai-Lewellen-Tye \\cite{Kawai:1985xq} relation\n\\begin{align}\n{\\mathcal C}_{\\text{sGR}}^{\\mu_1\\nu_1,\\mu_2\\nu_2}=-\\kappa^2 \\ii \\, e^{\\ii b\\cdot(k_1+k_2)}\\hat{\\delta}( p \\cdot(k_1+k_2)) \\frac{k_1\\cdot p \\, k_2\\cdot p}{k_1\\cdot k_2} \\bar A_{\\text{sQED}}^{\\mu_1\\mu_2}\n\\bar A_{\\text{sQED}}^{\\nu_1\\nu_2} \\,,\n\\end{align}\nwith the QED current given in Eq.\\eqref{2ptsQED}, after replacing $e^{2}\\to \\kappa^{2}\/4$.\nIn order to fuse the full current into the amplitude \\eqref{sGR-Compton}, we use physical polarization tensors $\\epsilon^{\\mu\\nu}_{\\pm \\pm}(k_{i}) = \\epsilon^{\\mu}_{\\pm}(k_{i})\\epsilon^{\\nu}_{\\pm}(k_{i})$ written as a product of null transverse photon polarizations. We set $k_{1}$ as incoming momentum and $k_{2}$ as outgoing and choose the rest frame of the worldline, i.e.,\n\\begin{equation}\np^{\\mu}=m u^{\\mu} = (m,0,0,0),\n\\quad k^{\\mu}_{1} =E(1,0,0,1), \\quad\nk^{\\mu}_{2} = E(1,\\sin\\theta,0,\\cos\\theta),\n\\end{equation} \n where $E$ is the energy of the graviton. Explicit polarization vectors $\\epsilon^{\\mu}_{\\pm}$ follow from the transversality and traceless conditions. Therefore, we can evaluate \\eqref{sGR-Compton} for the independent set of helicity configurations $(++),(+-)$. The scalar contributions are given by \n\\begin{equation}\n|{\\cal M}_{++}|_{_{0}} = |{\\cal M}_{--}|_{_{0}}= \\frac{\\kappa ^{2}m^{2}}{4}\\frac{\\cos^{4}\\frac{\\theta }{2}}{ \\sin^{2}\\frac{\\theta }{2} },\n\\hskip.4cm \n|{\\cal M}_{+-}|_{_{0}} = |{\\cal M}_{-+}|_{_{0}} = \\frac{\\kappa ^{2} m^{2}}{4} \\sin^{2}\\frac{\\theta }{2}\n\\,.\n\\end{equation}\nSimilarly, using the full current the linear in spin terms read \n\\begin{equation}\n|{\\cal M}_{++}|_{_{O({\\cal S})}} = |{\\cal M}_{++}|_{_{0}}\\Big|\t\t\t\ns\\cdot(k_{1}+k_{2})\\tan^{2}\\frac{\\theta}{2} +\\ii \\frac{k_{1}\\cdot {\\cal S}\\cdot k_{2}}{mE \\cos^{2}\\frac{\\theta}{2}}\n\\Big|,\n\\hskip.4cm |{\\cal M}_{+-}|_{_{O({\\cal S})}} = |{\\cal M}_{+-}|_{_{0}} \\, |s\\cdot (k_{1}-k_{2})| \\,, \n\\end{equation}\nwhile the quadratic in spin contributions can be recast in the suggestive way\n\\begin{equation}\n|{\\cal M}_{++}|_{_{O({\\cal S}^{2})}} = \\frac{1}{2} \\frac{\\hskip.4cm |{\\cal M}_{++}|^{2}_{_{O({\\cal S})}} }{ |{\\cal M}_{++}|_{_{0}}},\n\\hskip.4cm |{\\cal M}_{+-}|_{_{O({\\cal S}^{2})}} = \\frac{1}{2} \\frac{ \\hskip.4cm |{\\cal M}_{+-}|^{2}_{_{O({\\cal S})}} }{ |{\\cal M}_{+-}|_{_{0}}} \\,.\n\\end{equation}\nThe remaining helicity configurations can be obtained by replacing ${\\cal S}^{a b }\\to -{\\cal S}^{a b},s^{a}\\to -s^{a}$.\nThe above results are in agreement with \\cite{Saketh:2022wap} up to quadratic in spin\nupon matching with our conventions.\n\n\\section{Application to Hard Thermal loops}\n\\label{currents-and-HTLs}\nA nice application of the ideas presented in the past sections is the case of Hard Thermal Loops (HTLs). These are currents in the high temperature limit which can be resumed and incorporated into an effective theory known as HTL effective theory~\\cite{Braaten:1989mz, Frenkel:1989br, Braaten:1990az, Taylor:1990ia,\n\tFrenkel:1991ts}. It is well known that the high temperature regime is equivalent to a classical regime. Using a KMOC-like approach this was explicitly demonstrated in Ref.\\cite{delaCruz:2020cpc}, where HTLs were computed as the limit $\\hbar \\to 0$. Schematically HTL currents can be written as \n\\begin{align} \n\\Pi_n(k)=\\int \\mathrm{d}\\Phi (p) f(p_{0})\n \\bar A_n(p, k) \\, ,\n\\label{htl}\n\\end{align} \nwhere $f (p_{0})$ is a distribution function at equilibrium and $\\bar A_n(p, k)$ is the classical limit of the current in the regularized forward limit. The regularization is required since the same diagrams that contribute to the currents contribute to amplitudes so in general the forward limit is singular.\n Let $\\mathcal F$ be the set of all Feynman diagrams contributing to the current \\eqref{general-def}. Diagrammatically the regularization consists on dropping the set of all diagrams\nproducing zero momentum internal edges $\\mathcal X$ in the forward limit\\footnote{\n\tThese problematic diagrams appear e.g., Eq.\\eqref{current-classical-worldlined-QED} and \\eqref{current-scalar-GR}\n\tso the regularization should also be implemented in the worldline formalism.}. It is defined by\n\\begin{align}\t\nA_n (p,k):=\\sum_{G\\in \\mathcal F \\backslash \\mathcal X}\nd(G)\\, ,\\label{reg-current}\n\\end{align}\nwhere $d(G)$ is a rational expression of the form $N(G)\/D(G)$. In the forward limit $p_1=-p_2$ so momentum conservation becomes\n\\begin{align}\n\\sum_{i=1}^{n}k_{i} =0.\n\\label{momentum-conservation}\n\\end{align}\nThe classical limit of Eq.\\eqref{reg-current} is obtained through Eq.\\eqref{final-current-KMOC}. These currents have been considered in Refs.\\cite{delaCruz:2020cpc, delaCruz:2022nlj}. \n\nThe WQFT approach gives us a new way of obtaining these currents. In QED the equivalence between\n\\eqref{final-current-KMOC} and \\eqref{current-classical-worldlined-QED} implies that the $n-$point HTL can be read off from\n\\begin{equation}\n \\hat{\\delta} \\left(p\\cdot \\sum\\limits_{i=1}^n k_i \\right)\\frac{1}{2}{\\bar A}_n (p, k) = \\ii k_{1}^{2}\\ii k_{2}^{2}\\cdots \\ii k_{n}^{2} \\langle\nA_{\\mu_{1}}(k_{1})A_{\\mu_{2}}(k_{2})\\cdots A_{\\mu_{n}}(k_{n}) \\rangle_{\\text{WQFT}}, \n\\end{equation}\nwhere the regularization is understood in both sides. Since we are interested in $\\bar A_n(p,k)$ we will strip-off the Dirac-delta produced by WQFT.\nInserting\nthe RHS of this equation side into Eq.\\eqref{htl} gives and alternative worldline path integral representation of the HTL resumed current. \nA similar matching can be used to obtain HTLs in other theories. \n\n\nDiagrams that contribute at each order in perturbation can be determined from dimensional analysis.\nThe treatment of colour in the classical limit requires special care due to nontrivial cancellations between color and kinematics.\n In the worldline context this is translated into the dynamics of auxiliary variables which are present in the worldline action and give additional interactions. Their role is to encode non commutativity of color factors in general. The fact that classical color factors commute in the classical limit is recovered after integration over the auxiliary variables, a property that requires representations of the gauge group to be large\n (See Appendix \\ref{colors-appendix}). \n \n When classical color factors are recovered, thermal currents are obtained after phase-space integration over color. Phase space integration over classical color factor is defined by\n \\begin{align}\n \\mathrm{d} c:=& \\mathrm{d}^8 c\\, c_R \\delta(c^a c^b \\delta^{ab}-q_2 ) \\delta(d^{abc}c^a c^bc^c-q_3 ), \\label{DIPSC}\n \\end{align}\n %\n where $q_2$ and $q_3$ are Casimir invariants. The factor $c_R$ ensure that the color measure is normalized to unity and we have set the gauge group to be $SU(3)$. For bi-adjoint scalars we will take two copies of the phase-space integration measure.\n\n\\subsection{Scalars}\n\\label{Bi-adjoint}\nAs our first application let us consider scalar theories with cubic interactions. For instance, the bi-adjoint field theory Lagrangian is given by \\cite{Luna:2015paa,White:2016jzc}\n\\begin{align} \nS_{\\text {BA}}[\\varphi]=\\int\\mathrm{d}^4 x \\left[\\frac 12 \\partial_\\mu \\varphi^{a \\alpha} \\partial^\\mu \\varphi^{a \\alpha} \n-\\frac{m^2}{2} \\varphi^{a \\alpha}\\varphi^{a \\alpha} \n+\\frac{y}{3!} f^{abc} \\tilde f^{\\alpha \\beta \\gamma}\n\\varphi^{a \\alpha} \\varphi^{b \\beta} \\varphi^{c \\gamma}\\right] \\:, \n\\label{lag-phi3}\n\\end{align}\nwhere $m$ is the mass and $y$ the coupling constant. The bi-adjoint scalar field $\\varphi^{a\\alpha}$ transforms under the adjoint representation for each factor of its globally symmetry group $G \\times \\tilde G$. The Lie algebra for each factor has the form $[T^a, T^b]=f^{abc} T^c$ and the adjoint representation is given by its structure constants, i.e.\n$(T_A^a)^{b}_{\\ c}=- f^{abc}$. Throughout we use Greek indices for the group $\\tilde G$. We briefly outline the derivation of WFRs for this theory in Appendix \\ref{bi-adjoint-WQFT}. It\nis instructive to present the colorless and colorful cases separately to see how the appearance of color affect the final results. \n %\n\\subsubsection*{Cubic scalars}\nLet's consider the simplest example, where we have two soft scalars. Two equivalent diagrams with symmetry factor $\\frac 12$ are generated by the path integration over scalar fields. Hence it is enough to consider the one in Fig.\\ref{lo-gen} whose symmetry factor is unity. Using WFRs we obtain \n\\begin{align} \\label{tp1}\n \\raisebox{-.9cm}{\n\\lotop{$\\varphi(k_1)$}{$\\varphi(k_2)$}{black}}=\n \\frac{1}{2} \n \\bar A_2 (p,k_1)\n=& \\ii \n\\frac{1}{4}y^2 \\frac{k_1^2}{(k_1\\cdot p)^2}\\, ,\n\\end{align}\nwhich we can use to extract $A (p,k_1)$. \nHence using momentum conservation, relabeling the independent momentum $k_1=k$ \nwe obtain\n\\begin{align}\n\\Pi_2 (k)= \\frac{y^2}{2} \\int\\mathrm{d} \\Phi (p) f(p_0)\n \\frac{k^2}{(k\\cdot p)^2}.\n\\end{align}\nThe same procedure can be carried on in order to evaluate the three point thermal current. Notice however that a diagram analogous to Fig.\\ref{compton-amplitude} also contributes to the current but is singular in the forward limit so we discard it due to the regularization. The result now depends on permutations of the \ndiagrams in Fig.\\ref{nlo-gen}, which can be easily evaluated\n\\begin{align}\n\\bar A_3(k_1,k_2,k_3)= \\frac{y^3}{4} \n\\sum\\limits_{\\sigma \\in \\text{Cyclic}} \\frac{k_{\\sigma_1}\\cdot k_{\\sigma_2} k_{\\sigma_2}\\cdot k_{\\sigma_3}}{\\left(p\\cdot\n\tk_{\\sigma_1}\\right){}^2 \\left(p\\cdot k_{\\sigma_3}\\right){}^2},\n\\end{align}\nwhere the sum runs only over cyclic permutations of $\\{1,2,3\\}$. This equation is in agreement with Ref.\\cite{delaCruz:2022nlj} obtained from semi-classical kinetic theory.\n\n\\subsubsection*{Bi-adjoint}\nThe bi-adjoint scalar also generates Feynman rules which involve color-color fluctuations and color-matter fluctuations, as explained in Appendix \\ref{bi-adjoint-WQFT}. It receives the contributions from the diagrams\\footnote{In the colored case, color fluctuations are of the same order in coupling as those with matter. However upon restoring $\\hbar$ one must also take into account that color factorization brings an additional power of $\\hbar$. See Appendix \\ref{colors-appendix}.} with kinematic fluctuations of the worldlines\n %\n \\begin{align}\n\\raisebox{-0.7cm}{ \\lotop{}{}{black} } \n = \\ii \\frac{y^2}{4} c^{a_1} c^{a_2} \\tilde{c}^{\\alpha_1} \\tilde{c}^{\\alpha_2} \\frac{k_1\\cdot k_2}{(k_1\\cdot p)^2}\\,,\n \\end{align} \n and from the diagrams propagating color fluctuations\n \\begin{align}\n \\raisebox{-0.7cm}{ \\lotopcol{}{}{red}} +\n \\raisebox{-0.7cm}{ \\lotopcolCross{}{}{red}} \n =&\\, \\ii \\frac{y^2}{4} \\frac{ \\tilde{c}^{\\alpha_1} \\tilde{c}^{\\alpha_2}}{k_1\\cdot p}\\bar{u}\\cdot(T^{a_1}\\cdot T^{a_2}-T^{a_2} \\cdot T^{a_1} )\\cdot u\\, , \\label{cont2-bi}\\\\\n\\raisebox{-0.7cm}{ \\lotopcol{}{}{blue}}\n+\n \\raisebox{-0.7cm}{ \\lotopcolCross{}{}{blue}} \n=&\\, \\ii \\frac{y^2}{4} \\frac{c^{a_1} c ^{a_2}}{k_1\\cdot p}\\bar{v}\\cdot( \\tilde T^{\\alpha_1}\\cdot \\tilde T^{\\alpha_2}-\\tilde T^{\\alpha_2}\\cdot \\tilde T^{\\alpha_1})\\cdot v\\label{cont3-bi}\\, ,\n\\end{align}\nIt should be noticed how adding up the two topologies in each of Eq.\\eqref{cont2-bi} and \\eqref{cont3-bi} generates the structure constants of the Lie algebra \n\\begin{align}\n\\bar{u}\\cdot \\left( T^{a_1}\\cdot T^{a_2}-T^{a_2}T^{a_1} \\right)\\cdot u = f^{a_1 a_2 a_3}c^{a_3}\\, \n\\label{Lie-bracket}\n\\end{align}\nwhere $c^a=\\bar u\\cdot T^a \\cdot u$ and\nthe same for the tilded partner.\nThen, the current simplifies to \n\\begin{align}\n\\bar A_2^{a_1 \\alpha_1, a_2 \\alpha_2}= \\frac{y^2}{2}\\left( c^{a_1} c^{a_2} \\tilde c^{\\alpha_1} \\tilde c^{\\alpha_2} \\frac{ k_1^2}{(k_1 \\cdot p)^2}+\n\\frac{\\tilde c^{\\alpha_1} \\tilde c^{\\alpha_2}f^{a_1 a_2 a_3} c^{a_3} +c^{a_1} c^{a_2}\\tilde f^{\\alpha_1 \\alpha_2 \\alpha_3} \\tilde c^{\\alpha_3}}{k_1\\cdot p} \n\\right)\\, .\\end{align} \nThe phase-space integration over color can be done using the identities\n\\begin{align}\\label{psint}\n\\int \\mathrm{d} c\\ c^{a}=0, \\qquad \\int \\mathrm{d} c\\ c^{a} c^{b}= \\delta^{ab},\n\\end{align}\nwhich follow from Eq.\\eqref{DIPSC}. \nHence after some relabeling\n\\begin{align}\n\\Pi^ {a_1 \\alpha_1, a_2 \\alpha_2}(k)= \\delta^{a_1 a_2}\\delta^{\\alpha_1\\alpha_2 } \\frac{q_{2}^{2}y^{2}}{2}\\int\\mathrm{d} \\Phi(p) \\frac{ k^2}{(k \\cdot p)^2},\n \\end{align}\n %\nwhich \n is in agreement with kinetic theory of Ref.\\cite{delaCruz:2022nlj}.\n\n\\subsection{Gauge and gravity}\n\\label{gauge}\nAs our final example, we \nmove directly to scalar QCD since \n scalar QED HTLs can be recovered \nfrom the off-shell currents in \nSection \\ref{gauge-examples} using momentum conservation. In QED no singular diagrams arise in the forward limit. \n For instance, the 5-pt HTL it is straightforward to obtain from Eq.\\eqref{eq-5-pt} and Eq.\\eqref{momentum-conservation}. \n\nLet us consider the 3-point HTL. \nFrom the WFRs in Appendix \n\\ref{colors-appendix} there are two\ndiagrams we need to calculate. For example, \nfor the permutation $\\sigma=(1,2,3)$ for the external gluons one has \n\\begin{align}\n\\raisebox{-4mm}{ \\nlotopC } &= g^{3}\\ \\bar u\\cdot T^{a_{1}}\\cdot T^{a_{3}}u \\ c^{a_{2}}\n\\frac{p^{\\mu_{3} }}{p\\cdot k_{3} }\\bar A_{\\text{sQED}}^{\\mu_{2}\\mu_{1}} (k_{2},k_{1})\\,, \n\\\\\n\\raisebox{-4mm}{ \\nlotopD } &= -g^{3}\\ \\bar u\\cdot T^{a_{3}} \\cdot T^{a_{1}}u\\ c^{a_{2}}\n\\frac{p^{\\mu_{3} }}{p\\cdot k_{3} }\\bar A_{\\text{sQED}}^{\\mu_{2}\\mu_{1}} (k_{2},k_{1})\\,,\n\\end{align}\nwhich generate the $SU(N)$ structure constants as in Eq.\\eqref{Lie-bracket}. Performing the phase space integration \\eqref{psint} and summing over all permutations\n the final answer can be written as \n\\begin{equation}\n\\bar A^{\\mu_{1} \\mu_{2} \\mu_{3}}_{a_{1} a_{2} a_{3}}=- 2g^{3} \\sum_{\\sigma \\in S_3}\nf^{a_{\\sigma_1 } a_{\\sigma_2} a_{\\sigma_3}} \n\\frac{p^{\\mu_{\\sigma_3}}}{p\\cdot k_{\\sigma_3}} \\bar A_{\\text{sQED}}^{\\mu_{\\sigma_2}\\mu_{\\sigma_1}} (k_{\\sigma_2},k_{\\sigma_1})\\label{current-3pt}\n\\end{equation}\nwhere $S_3$ is the set of all permutations of $\\{1,2,3\\}$. \nThe above result satisfies the identity \n\\begin{equation}\n k_{3\\mu_{3}} \\, {\\bar A}^{\\mu_{1} \\mu_{2} \\mu_{3}}_{a_{1} a_{2} a_{3}} = 2g^{3} f^{a_{1} a_{2} a_{3}} \\left(\n\\bar A_{\\text{sQED}}^{\\mu_{1}\\mu_{2}} (k_{1},-k_{1}) -\\bar A_{\\text{sQED}}^{\\mu_{1}\\mu_{2}} (k_{2},-k_{2})\n\\right)\n\\end{equation}\nand can be\nstraightforwardly brought into the form given in Ref.\\cite{delaCruz:2020cpc}. The form of Eq.\\eqref{current-3pt} shows the direct connection between QED and QCD in the high temperature regime.\n\nWe have have checked that the WQFT approach reproduces the higher-point \nexamples considered in Refs.\\cite{delaCruz:2020cpc, delaCruz:2022nlj}\nincluding the \ngravitational case. It is then safe to conjecture that the remarkable simple expression\n\\begin{equation}\n k_{1}^{2}k_{2}^{2}\\cdots k_{n}^{2} \\langle\nA^{I_1}(k_{1})A^{I_{2}}(k_{2})\\cdots A^{I_{n}}(k_{n}) \\rangle_{\\text{WQFT}}\n\\end{equation}\nencodes the resumed HTL expansion for any theory where scalars interact with other scalar, gauge bosons or gravitons. \n\n\\section{Conclusions}\n\\label{conclusions}\nWe have exposed a relation between the WQFT path integral and the classical limit of an off-shell current. The latter\nobtained through dressing up an off-shell current with first quantized coherent states and its classical limit obtained through a KMOC-like\nprocedure. Using a localized wave-packet to\ncompute classical currents is equivalent to pick up the background expansion $x^{\\mu} = b^{\\mu }+p^{\\mu}\\tau +q^{\\mu}(\\tau)$ and setting $\\tau\\in\\, (-\\infty,\\infty)$ in the worldline action as we have shown in Sec.\\ref{off-shell-classical}.\nWe then constructed the path integral representation of the Green functions associated with the off-shell current and found a link to the worldline theory in the classical limit, thus \nrecasting the current as an expectation value of soft fields inside a worldline path integral. \n\nWe have applied our methods\nin various settings. In QED we have tested the validity of our approach up to 7-point, while in gravity we have verified that the \n${\\cal N}=2$ susy model correctly reproduces the classical Compton amplitude up to quadrupole finding agreement with Ref.\\cite{Saketh:2022wap}. For colored scalars and QCD, we have applied our methods to obtain HTLs, which are related to the classical limit of off-shell currents \\cite{delaCruz:2020cpc}. \nIt is straightforward to extend our approach to include more particles either massive or massless. For instance one could easily include photons using the dressed propagator of Ref.\\cite{Bastianelli:2021nbs}, while on the KMOC side massless particles were studied in Ref.\\cite{Cristofoli:2021vyo}. \n \nOur work shows how to map computations of the classical\ncontributions of the off-shell (amplitude-like) current and the\ncalculation based \non worldline path integration, thus providing a route to test future WQFT developments. \nFor instance a possible\napplication of our currents would be the case of \nmassive higher-spin particles, where one could compare the off-shell current in the classical limit and the output produced by WQFT. \n \n\n\nOff-shell currents are usually computed recursively in high-energy-physics applications so it would be interesting to recursively compute currents WQFT too. Hints of their recursive structure have already appeared in Ref.\\cite{Jakobsen:2021zvh}. Although we have considered only tree-level currents our construction can be relaxed to include loops, which can be used in the calculation of loop-level\ncorrections to the background metric in general relativity \\cite{DOnofrio:2022cvn,Mougiakakos:2020laz,Jakobsen:2020ksu}. On the other hand, cuts that contribute classically \n might arise from sewing tree-level currents as it is usual in the generalized unitarity method \n \\cite{Bern:1994cg, Bern:1994zx}. \n \n\\addsec{Acknowledgements}\nFC would like to thank Canxin Shi for helpful discussions. \n LDLC acknowledges financial support from the Open Physics Hub at the Physics and Astronomy Department in Bologna. His work is also supported by \nthe European Research Council, under grant ERC-AdG-885414. \n Some of the \ncalculations in this paper were done with Feyncalc \\cite{Mertig:1990an, Shtabovenko:2016sxi, Shtabovenko:2020gxv}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\subsection{Motivation}\n\n\nOne of the first phase transition results for random graphs is the celebrated result of Erd\\H{o}s and R\\'enyi~\\cite{ErdosRenyi60} on the emergence of a \\emph{giant component} of linear\norder when the number of edges passes $\\frac{n}{2}$, or from the viewpoint that is now more common,\nwhen the edge probability passes $\\frac{1}{n}$. This result has since been strengthened and generalised\nin a number of directions. In particular in hypergraphs it has been extended to vertex-components\n(e.g.~\\cite{BCOK10,BCOK14,BollobasRiordan12,BollobasRiordan17,KaronskiLuczak02,Poole15,SPS85})\nas well as to high-order components (e.g.~\\cite{CFDGK20,CKK18,CKK19,CKP18}).\n\n\nThe $k$-core of a graph $G$, defined as the maximal subgraph of minimum degree at least~$k$, has been studied extensively in the literature (e.g.~\\cite{Cooper2004, JansonLuczak07, kimcore, LuczakCore1991, PittelSpencerWormald96}). \nIn random graphs, the $k$-core may be seen as a natural generalisation of the largest component: \nin the case $k=2$, whp\\footnote{short for \\emph{with high probability}, i.e.\\ with probability tending to one as the number of vertices $n\\to \\infty$.}\na linear-sized $2$-core emerges at the same time as the giant component, and indeed\nlies almost entirely within the giant component, while for $k\\ge 3$, whp the $k$-core\nis identical to the largest $k$-connected subgraph~\\cite{LuczakCore1991,Luczak92}. \nIn~\\cite{LuczakCore1991} {\\L}uczak estimated the order of (i.e.\\ the number of vertices in)\nthe $k$-core of $G(n,p)$ and the asymptotic probability that the $k$-core is $k$-connected.\n{\\L}uczak also showed in~\\cite{Luczak92} that in the random graph process, in which edges are added to an empty graph one by one in\na uniformly random order, whp\\ at the moment the $k$-core first becomes non-empty, its order\nis already linear in~$n$. \nA crucial milestone was achieved by Pittel, Spencer and Wormald~\\cite{PittelSpencerWormald96},\nwho for $k\\geq 3$ determined the threshold probability at which the non-empty $k$-core appears whp\\ and\ndetermined its asymptotic order and size (i.e.\\ number of edges).\nThis was strengthened by Janson and Luczak~\\cite{JansonLuczak08}, who proved a bivariate\ncentral limit theorem for the order and size of the $k$-core.\nCain and Wormald~\\cite{CainWormald06} determined the asymptotic distribution\nof vertex degrees within the $k$-core.\nFurther research has focussed for example on the robustness of the core against edge deletion~\\cite{Sato14}\nand how quickly the peeling process arrives at the core~\\cite{AchlioptasMolloy15,Gao18,GaoMolloy18,JMT14}.\nThere are many more results in the literature for cores in random graphs, see e.g. \\cite{Cooper2004, JansonLuczak07, kimcore}. \n\n\nPaths and cycles in random graphs have been investigated at least since 1979 by de la Vega~\\cite{delavega79} and somewhat later by Ajtai, Koml\\'os,\nand Szemer\\'edi~\\cite{AjtaiKomlosSzemeredi81}.\nRegarding the length of the longest path in the random graph $G(n,p)$,\na standard ``sprinkling'' argument (see Lemma~\\ref{lem:standardsprinkling} with $r=2$)\nshows that in the supercritical regime the length of the longest path and cycle are very similar.\nThus it follows from the results of {\\L}uczak~\\cite{Luczak91} on the length of the longest cycle\nthat for $\\varepsilon=\\varepsilon(n)=o(1)$ and $p=\\frac{1+\\varepsilon}{n}$ (i.e.\\ shortly after the phase transition),\nunder the assumption $\\varepsilon^5n\\to\\infty$ whp\\ the longest path has length $\\Theta(\\varepsilon^2 n)$, where explicit constants can be given.\nThe best-known upper bounds derive from a careful analysis of the $2$-core\nand the simple observation that any cycle must lie within the $2$-core.\n\n\n\n \nThere are many different ways of generalising the concept of a $k$-core to hypergraphs; some results for these cores can be found in e.g. Molloy~\\cite{Molloy2005} and Kim~\\cite{kimcore}. However,\nin the case $k=2$,\nall $k$-cores which have been studied so far do not fully capture the nice connection between the $2$-core and cycles in graphs.\nIn~\\cite{Molloy2005} Molloy determined the threshold for the appearance of a non-trivial $k$-core\n(in that paper defined as a subhypergraph where every vertex has degree at least $k$)\nin the $r$-uniform binomial random hypergraph $H^r(n,p)$ for all $r,k\\geq 2$ such that $r+k\\geq 5$.\nThe proof relied on a clever heuristic argument which was first introduced by Pittel, Spencer and Wormald in~\\cite{PittelSpencerWormald96} and has been adapted by many other authors, see e.g.~\\cite{JansonLuczak07, kimcore,Riordan08,skubch2015core}. It turns out that the proofs in~\\cite{Molloy2005} can be extended to a wide range of core-type structures.\nIn the case $k=2$, Dembo and Montanari~\\cite{DemboMontanari08} strengthened this by determining\nthe width of, and examining the behaviour within, the critical window.\n\n\nOne of the most natural concepts of paths and cycles in hypergraphs\nis \\emph{loose paths} and \\emph{loose cycles} (see Definition~\\ref{def:loosecycle}).\nA special case of a recent result of Cooley, Garbe, Hng, Kang, Sanhueza-Matamala and Zalla~\\cite{cooley2020longest}\nshows that the length of the longest loose path in an $r$-uniform binomial random hypergraph undergoes\na phase transition from logarithmic length to linear, and they also determined the critical threshold,\nas well as proving upper and lower bounds on the length in the subcritical and supercritical ranges.\n\nInspired by the substantial body of research on loose cycles, in this paper we introduce the\n\\emph{loose core} (see Definition~\\ref{def:loosecore}), a structure which does indeed capture the connection between cores and cycles in hypergraphs.\nOur first main result concerns the degree distribution of vertices in the loose core (see Theorem~\\ref{thm:mainresultdegrees}). In fact we prove a stronger result regarding degree distributions of both vertices and edges (see Theorem~\\ref{thm:factor:reducedcoredegs}).\nAs a consequence we can deduce both the asymptotic numbers of vertices and edges in the loose core (see Theorem~\\ref{thm:mainresultorder}) and an improved upper bound on the length of the longest loose cycle in an $r$-uniform binomial random hypergraph (see Theorem~\\ref{thm:mainresultcycle}). \n\nBefore stating our main results, in the next section we introduce some definitions and notations which we will use throughout the paper.\n\n\n\\subsection{Setup}\\label{def:basicdefinitions}\nGiven a natural number $r\\geq 3$,\nan \\emph{$r$-uniform hypergraph} consists of a vertex set $V$ and an edge set\n$E\\subset\\binom{V}{r}$, where $\\binom{V}{r}$ denotes the set of all $r$-element subsets of $V$. Let $H^r(n,p)$ denote the \\emph{$r$-uniform binomial random hypergraph}\non vertex set $[n]$ in which each set of $r$ distinct vertices forms an edge with probability $p$ independently. For any positive integer $k$ we write $[k]\\coloneqq\\{1,\\ldots,k\\}$ and $\\ [k]_0\\coloneqq\\{0,\\ldots,k\\}$.\nWe also include $0$ in the natural numbers, so we write $\\mathbb{N}=\\{0,1,\\ldots\\}$ and $\\mathbb{N}_{\\geq k}\\coloneqq\\{k,k+1,\\ldots\\}$.\nThroughout the paper, unless otherwise stated any asymptotics are taken as $n\\to\\infty$. \nIn particular, we use the standard\nLandau notations $o(\\cdot)$, $O(\\cdot),\\Theta(\\cdot),\\omega(\\cdot)$\nwith respect to these asymptotics.\n\nThe loose core will be defined in terms of two parameters, namely the standard notion of (vertex-)degree\nand a notion we call the connection number.\n\n\\begin{definition}\nLet $H$ be an $r$-uniform hypergraph. Let $d_H(v)$ be the \\emph{degree} of a vertex $v$ in $H$ (i.e.\\ the number of edges which contain it) and let $\\delta(H)$ denote the \\emph{minimum (vertex-)degree} of $H$, i.e.\\ the smallest degree of any vertex of $H$. For any edge $e\\in E(H)$, define the \\textit{connection number} $\\kappa(e)\\in[r]_0$ of $e$ as \\[\\kappa(e)=\\kappa_H(e)\\coloneqq|\\{v\\in e: d_H(v)\\geq 2\\}|\\] and let $\\kappa(H)\\coloneqq\\min\\limits_{e\\in E(H)}\\kappa(e)$.\n\\end{definition}\nWe are now ready to define the loose core.\n\\begin{definition}[Loose core]\\label{def:loosecore}\nThe \\textit{loose core} of an $r$-uniform hypergraph $H$ is the maximal subhypergraph $H'$ of $H$ such that \n\\begin{enumerate}[label=\\textnormal{\\textbf{(C\\arabic*)}}]\n \\item\\label{item:loosecorefirstcond} $\\delta(H')\\geq 1$, \n \\item\\label{item:loosecoresecondcond} $\\kappa(H')\\geq 2$.\n\\end{enumerate}\nIf such a subhypergraph does not exist, then we define the loose core to be the empty hypergraph (i.e.\\ the hypergraph with no vertices and no edges).\n\\end{definition}\nNote that the loose core is unique, since the union of two hypergraphs each with properties \\ref{item:loosecorefirstcond} and \\ref{item:loosecoresecondcond} again has these properties. The first condition in Definition~\\ref{def:loosecore} simply states that the loose core contains no isolated vertices and the second condition specifies how edges are connected to each other in the loose core. Note that for $r\\ge 3$ the loose core might contain vertices of degree $1$, in contrast to the graph case. For $r=2$, Definition~\\ref{def:loosecore} coincides with the $2$-core of a graph.\n\nOur motivation to study loose cores arises from the study of loose cycles in hypergraphs which are closely related to loose paths.\n\\begin{definition}[Loose path\/cycle]\\label{def:loosecycle}\nA \\textit{loose path of length $\\ell$} in an $r$-uniform hypergraph is a sequence of distinct vertices $v_1,\\ldots,v_{\\ell(r-1)+1}$ and a sequence of edges $e_1,\\ldots,e_{\\ell}$, where $e_i=\\{v_{(i-1)(r-1)+1},\\ldots,v_{(i-1)(r-1)+r}\\}$ for $i\\in[\\ell]$. A \\textit{loose cycle of length $\\ell$} in an $r$-uniform hypergraph is defined similarly except that $v_{\\ell(r-1)+1}=v_1$ (and otherwise all vertices are distinct).\n\\end{definition}\n\nNote that for $i \\in [\\ell-1]$ we have $e_i \\cap e_{i+1}=\\{v_{i(r-1)+1}\\}$ (and in the case of a loose cycle,\n$e_\\ell \\cap e_1 = \\{v_1\\}$), so in particular two consecutive edges\nintersect in precisely one vertex.\nObserve that a loose cycle satisfies conditions~\\ref{item:loosecorefirstcond} and~\\ref{item:loosecoresecondcond} of a loose core (Definition~\\ref{def:loosecore}) and hence it must be contained in the maximal subhypergraph with these properties,\ni.e.\\ in the loose core.\n\nWe will now define various parameters which will occur often in this paper. Some of these\ndefinitions may seem arbitrary and unmotivated initially, but their meaning will become clearer over the course of the paper.\nGiven $d>0$, consider a sequence $(d_n)_{n \\in \\mathbb{N}}$ of real numbers such that $d_n\\to d$.\nThen for\n$r\\in\\mathbb{N}_{\\geq 3}$ and $n\\in\\mathbb{N}$, set\n\\[\np=p(r,n)\\coloneqq\\frac{d_n}{\\binom{n-1}{r-1}}, \\qquad d^*=d^*(r)\\coloneqq\\frac{1}{r-1}.\n\\]\nIn addition we define a function $F:[0,\\infty) \\to\\mathbb{R}$ by setting\n\\begin{equation*}\\label{eq:gfunctiondefinition}\n F(x)=F_{r,d}(x)\\coloneqq\\exp{\\left(-d\\left(1-x^{r-1}\\right)\\right)}\n\\end{equation*}\nand let $\\rho_*=\\rho_*(r,d)$ be the largest solution\\footnote{$\\rho_*$ is well-defined since $0$ is certainly a solution and the set of solutions is closed by continuity.} of the fixed-point equation\n\\begin{equation}\\label{eq:fixedpointequation}\n 1-\\rho=F(1-\\rho). \n\\end{equation}\nSince the function $F$ is dependent on $d$, so too are the solutions to this equation.\nIt turns out that $d^*$ is a threshold at which the solution set changes its behaviour from only containing the trivial solution $0$ to containing a unique positive solution (see Claim~\\ref{claim:behaviouroffixedpointsol}). \n\n\nWe define\n\\begin{equation}\\label{eq:relatingtherhos}\n\\hat \\rho_*=\\hat \\rho_*(r,d)\\coloneqq1-(1-\\rho_*)^{r-1}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:eta}\n\\eta=\\eta(r,d)\\coloneqq1-\\frac{(r-1)\\rho_*(1-\\rho_*)^{r-2}}{\\hat \\rho_*}.\n\\end{equation}\n Furthermore let \n\\begin{equation*}\\label{eq:alphadefinition}\n \\alpha=\\alpha(r,d)\\coloneqq\\rho_*\\left(1-d(r-1)(1-\\rho_*)^{r-1}\\right),\n\\end{equation*}\n\\begin{equation}\\label{eq:definitionofeta}\n\\beta=\\beta(r,d)\\coloneqq\\frac{d}{r}\\left(1-(1-\\rho_*)^ r-r\\rho_*(1-\\rho_*)^ {r-1}\\right),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:definitionofgamma}\n \\gamma=\\gamma(r,d)\\coloneqq1-\\exp(-d\\hat \\rho_*)-d\\hat \\rho_* \\exp(-d\\hat \\rho_*).\n\\end{equation}\n\n\n\\subsection{Main results: loose cores and cycles in hypergraphs}\\label{sec:mainresults}\n\n\nFor any $j\\in\\mathbb{N}_{\\ge 1}$, let $\\numvsdeg{j}{C_H}$ be the number of vertices of $H= H^r(n,p)$\nwith degree~$j$ in the loose core $C_H$ of $H$\nand let \n\\[\\mu_j\\coloneqq\\numvsdeg{j}{C_H}\\cdot n^{-1}.\\]\nLet $\\numvs{C_H} = \\sum_{j \\ge 1}\\numvsdeg{j}{C_H}$ denote the number of vertices and $\\numedg{C_H}$ the number of edges in the loose core $C_H$ in $H$.\nWe also define $\\numvsdeg{0}{C_H}$ to be the number of vertices of $H$ which are not in the loose core of $H$ (so $v_0(C_H)=n-v(C_H)$),\nand $\\mu_0\\coloneqq \\numvsdeg{0}{C_H}\\cdot n^{-1}$. (Observe that this notation is consistent if, with a slight abuse of terminology,\nwe view vertices which are not in the loose core as having degree $0$ in the loose core.) We interpret a $\\mathrm{Po}(0)$ variable as being deterministically $0$. \n\nOur first main result describes the asymptotic degree distribution of vertices in the loose core $C_H$ of $H= H^r(n,p)$.\n\\begin{theorem}\\label{thm:mainresultdegrees}\nLet $r,d,p,\\hat \\rho_*$ and $\\eta$ be as in Section~\\ref{def:basicdefinitions} and let $H=H^r(n,p)$. Let $Y$ be a random variable with distribution $\\mathrm{Po}(d\\hat \\rho_*)$ and define\n\\[\n Z \\coloneqq \\bigg\\{\\begin{array}{lr}\n Y & \\text{if } Y\\neq 1,\\\\\n \\mathrm{Ber}\\left(\\eta\\right) & \\text{if } Y=1.\n \\end{array}\\] \nThen there exists $\\varepsilon = \\varepsilon(n) = o(1)$ such that whp\\ for any constant $j\\in\\mathbb{N}$ we have\n\\[\\mu_j=\\prob(Z=j)\\pm \\varepsilon.\\]\n\\end{theorem}\nOur second main result describes the asymptotic numbers of vertices and edges in the loose core $C_H$ of $H= H^r(n,p)$.\n\n\\begin{theorem}\\label{thm:mainresultorder}\nLet $r,p, \\alpha$ and $\\beta$ be as in Section~\\ref{def:basicdefinitions} and let $H=H^r(n,p)$.\nThen whp\n\t\t\\[\\numvs{C_H}=(\\alpha+o(1)) n\\]\n\t\tand\n\t\t\\[\\numedg{C_H}=(\\beta+o(1))n.\\]\n\\end{theorem}\n\nBy analysing the loose core we obtain an upper bound on the length of the longest loose cycle in $H^r(n,p)$.\n\n\\begin{theorem}\\label{thm:mainresultcycle} \nLet $r,p, \\beta$ and $\\gamma$ be as in Section~\\ref{def:basicdefinitions} and let $H=H^r(n,p)$.\nLet $L_C$ be the length of the longest loose cycle in $H$. Then whp\n\\[L_C\\leq \\left(\\min\\{\\beta,\\gamma\\}+o(1)\\right)\\cdot n.\\]\n\\end{theorem}\n\nIn fact, a standard ``sprinkling'' argument shows that whp the longest loose path in $H^r(n,p)$ is not significantly longer than the longest loose cycle and therefore we obtain the following corollary.\n\n\\begin{corollary}\\label{cor:upperboundlongestpath} \nLet $r,p, \\beta$ and $\\gamma$ be as in Section~\\ref{def:basicdefinitions} and let $H=H^r(n,p)$. \nLet $L_P$ be the length of the longest loose path in $H$. Then whp\n\\[L_P\\leq \\left(\\min\\{\\beta,\\gamma\\}+o(1)\\right)\\cdot n.\\]\n\\end{corollary}\n\nAs mentioned previously, Claim~\\ref{claim:behaviouroffixedpointsol} will state that $d^*=\\frac{1}{r-1}$ is a threshold at which the solution set of~\\eqref{eq:fixedpointequation} changes its behaviour from only containing $0$ to containing a unique positive solution. Together with Theorem~\\ref{thm:mainresultcycle} and Corollary~\\ref{cor:upperboundlongestpath} (and recalling the definitions of $\\beta,\\gamma$\nin~\\eqref{eq:definitionofeta} and~\\eqref{eq:definitionofgamma}) this implies that $d^*=\\frac{1}{r-1}$ is a threshold for the existence of a loose path\/cycle of linear order,\nand it is interesting in particular to examine the behaviour shortly after the phase transition.\n\\begin{corollary}\\label{cor:shortlyafterphasetransition}\nLet $r\\in\\mathbb{N}_{\\geq 3}$, let $\\varepsilon>0$ be constant and let $p=\\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$. Let $L_C$ and $L_P$ be the length of the longest loose cycle and the longest loose path in $H^r(n,p)$. Then whp\n\\[L_C\\leq L_P+1\\leq \\left(\\frac{2\\varepsilon^2}{(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n.\\]\n\\end{corollary}\n\nIn other words, we have an upper bound on $L_C$ and $L_P$ in the barely supercritical regime.\nFor a corresponding lower bound, we will quote (a special case of) a result from~\\cite{cooley2020longest}\n(which we later state formally as Theorem~\\ref{thm:pathsresult}) which gives a lower bound on $L_P$.\nBy applying the sprinkling argument again we also obtain a lower bound on $L_C$,\nand together with Corollary~\\ref{cor:shortlyafterphasetransition} we obtain the following.\n\n\n\\begin{theorem}\\label{thm:bestknownresultcycles}\nLet $r\\in\\mathbb{N}_{\\geq 3}$, let $\\varepsilon>0$ be constant and let $p=\\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$. Let $L_C$ and $L_P$ be the length of the longest loose cycle and the longest loose path in $H^r(n,p)$. Then whp\\ \n\\[\n\\left(\\frac{\\varepsilon^2 }{4(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n\\leq L_C\\leq L_P+1\\leq\\left(\\frac{2\\varepsilon^2}{(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n .\n\\]\n\\end{theorem}\nTheorem~\\ref{thm:bestknownresultcycles} provides the best known upper and lower bounds on $L_P,L_C$ in the regime when\n$p= \\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$, \nbut there is a multiplicative factor of~$8$ between these two bounds and the correct asymptotic value is still unknown.\nIndeed it is not even clear that the random variables $L_P,L_C$ are concentrated around a single value.\n\nWe note that even for cycles in $G(n,p)$ in the barely supercritical regime the correct asymptotic value is not known\ndespite considerable efforts in this direction. In particular, the best known bounds\nwhen $0<\\varepsilon=\\varepsilon(n)=o(1)$ satisfies $\\varepsilon^3n\\to\\infty$ and $p=\\frac{1+\\varepsilon}{n}$\nare due to {\\L}uczak~\\cite{Luczak91} (lower bound) and Kemkes and Wormald~\\cite{KemkesWormald13} (upper bound),\nand state that whp\\ the length $L_C$ of the longest cycle satisfies\n\\[\n\\left(\\frac{4}{3}+o(1)\\right)\\varepsilon^2n\\leq L_C\\leq(1.7395+o(1))\\varepsilon^2n.\n\\]\n\nThe proofs of all the results of this section appear in Section~\\ref{sec:proofofmainresults} as a consequence of a single, more general result (Theorem~\\ref{thm:factor:reducedcoredegs}).\n\n\n\n\n\n\n\\subsection{Key proof techniques}\nIn order to prove our main results, we transfer the problem from $H^r(n,p)$ to the \\emph{factor graph} $G\\coloneqq G(H^r(n,p))$\nwhich will be formally defined in Section~\\ref{sec:factorgraphs}. In the factor graph we define the \\emph{reduced core} $R_G$,\nwhich is closely related to the $2$-core of $G$ and from which we can reconstruct the loose core of $H^r(n,p)$, but which is easier to analyse. We use a peeling process (Definition~\\ref{def:peeling}) and an auxiliary algorithm called $\\coredetector $ to determine the asymptotic proportion of variable and factor nodes of $G$ with fixed degree in the reduced core (Theorem~\\ref{thm:factor:reducedcoredegs}). We also need martingale techniques, in particular an Azuma-Hoeffding inequality and an associated vertex-exposure martingale to show concentration of the numbers of vertices and edges of fixed degree around the respective expectations. \n\\subsection{Paper overview}\nThe rest of the paper is structured as follows. \n\nIn Section~\\ref{sec:prelim} we set basic notation and state some standard probabilistic lemmas which we will use later.\nIn Section~\\ref{sec:factorgraphs} we switch our focus to factor graphs,\ndefine the reduced core and state Theorem~\\ref{thm:factor:reducedcoredegs}\nwhich describes degree distributions in the reduced core and which\nimplies all of our main results, as we prove in Section~\\ref{sec:proofofmainresults}.\n\nSubsequently, Section~\\ref{sec:peeling} describes a standard peeling process to obtain the reduced core and contains two main lemmas\nwhich together imply Theorem~\\ref{thm:factor:reducedcoredegs}.\nThe first of these (Lemma~\\ref{lem:mainlemma1}) describes the degree distribution after a sufficiently large number of steps of the peeling process,\nand will be proved in Section~\\ref{sec:algorithm}.\nThe second main lemma (Lemma~\\ref{lem:mainlemma2}) states that subsequently, very few further vertices will be deleted in the remainder of the peeling process,\nand therefore this degree distribution is also a good approximation for the degree distribution in the reduced core.\nLemma~\\ref{lem:mainlemma2} will be proved in Section~\\ref{sec:prooflowerbound}.\n\nIn Section~\\ref{sec:concluding}, we conclude with some discussion and open questions.\nWe omit from the main body of the paper many proofs which simply involve technical calculations\nor standard applications of common methods,\nbut include them in the appendices for completeness.\nAppendix~\\ref{app:analysisfixedpoint} contains an analysis of the fixed-point equation~\\eqref{eq:fixedpointequation},\nwhile Appendix~\\ref{app:problemmas} contains the proofs of some basic probabilistic lemmas which are needed throughout the paper.\nFinally, Appendix~\\ref{app:eventE} and Appendix~\\ref{sec:proofofuniformity} constitute the proofs of Lemma~\\ref{lem:eventE}\nand Lemma~\\ref{claim:uniformity}, respectively.\n\n\\section{Preliminaries and Notation}\\label{sec:prelim}\n\nFor the rest of the paper, $r\\in\\mathbb{N}_{\\geq 3}$ and $d>0$ will be fixed.\nIn particular, we consider these to be constant, so if we say, for example,\nthat $x = O(n)$, we mean that there exists a constant $C=C(r,d)$\nsuch that $x \\le Cn$.\nBy the notation $x=a\\pm b$ we mean that $a-b\\leq x\\leq a+b$.\nSimilarly, the notation $x = (a\\pm b)c$ means that $(a-b)c \\le x \\le (a+b)c$.\nWe will omit floors and ceilings whenever these do not significantly affect the argument. \\\\\n\nAs mentioned in Section~\\ref{def:basicdefinitions}, the solution set of the fixed-point equation~\\eqref{eq:fixedpointequation}\nchanges its behaviour at $d=d^*$. More precisely we have the following. \n\n\\begin{claim}\\label{claim:behaviouroffixedpointsol}\\mbox{ }\n\\begin{enumerate}[label=\\textnormal{\\textbf{(F\\arabic*)}}]\n\\item If $dd^*$, then there is a unique positive solution to~\\eqref{eq:fixedpointequation}.\n\\end{enumerate}\n\\end{claim}\nWe defer the (rather technical) proof of this claim to Appendix~\\ref{app:analysisfixedpoint}.\n\nFurthermore we will often use the following alternative relation between $\\rho_*$ and $\\hat \\rho_*$.\n\\begin{equation}\\label{eq:relatingrhoswithexpfunction}\n1-\\rho_*\\numeq{\\eqref{eq:fixedpointequation}}F(1-\\rho_*)=\\exp\\left(-d\\left(1-(1-\\rho_*)^ {r-1}\\right)\\right)\\numeq{\\eqref{eq:relatingtherhos}}\\exp(-d\\hat \\rho_*).\n\\end{equation}\n\n\\subsection{Large deviation bounds}\n\nIn this section, we collect some standard large deviation results which will be needed later. We will use the following Chernoff bound\n(see e.g.~\\cite[Theorem 2.1]{JansonLuczakRucinskiBook}).\n\\begin{lemma}[Chernoff]\\label{lem:chernoffbounds}\n\tIf $X\\sim \\mathrm{Bi}(N,p)$, then for any $s> 0$\n\t\\begin{equation*} \\label{eqn:chernoffstd}\n\t\\mathbb{P}(|X-Np|\\geq s)\\leq 2\\cdot\\exp\\left(-\\frac{s^2}{2\\left(Np+\\frac{s}{3}\\right)}\\right).\n\t\\end{equation*}\n\t\n\\end{lemma}\nThis bound is less precise than the form in~\\cite{JansonLuczakRucinskiBook} since we have combined the upper and lower tail bounds for simplicity.\nWe will also need an Azuma-Hoeffding inequality---the following lemma is taken from~\\cite{JansonLuczakRucinskiBook}, Theorem $2.25$.\n\\begin{lemma}[Azuma-Hoeffding]\\label{lem:Azuma}\nLet $X$ be a real-valued random variable.\nIf $(X_i)_{1\\leq i\\leq n}$ is a martingale with $X_n=X$ and $X_0=\\expec(X)$,\nand for every $1\\leq i\\leq n$ there exists a constant $c_i>0$ such that\n\\[|X_i-X_{i-1}|\\leq c_i\\]then, for any $s>0$,\n\\begin{align}\n \\prob(|X-\\expec(X)|\\geq s) & \\leq 2\\cdot\\exp\\left(-\\frac{s^2}{2\\sum_{i=1}^nc_i^2}\\right).\\nonumber\n\\end{align}\n\\end{lemma}\nAgain, the bound in~\\cite{JansonLuczakRucinskiBook} is a bit stronger than what we state here since for simplicity we have combined upper and lower tail bounds.\n\n\\section{Factor graphs}\\label{sec:factorgraphs}\n\nThere is a natural representation of a hypergraph as a bipartite graph known as a \\emph{factor graph},\nwhich is a well-known concept in literature (see e.g.~\\cite{MezardMontanariBook}).\n\n\\begin{definition}[Factor graph]\nGiven a hypergraph $H$, the \\emph{factor graph} $G=G(H)$ of $H$ is a bipartite graph on vertex classes $\\mathcal{V}\\coloneqq V(H)$ and $\\mathcal{F}\\coloneqq E(H)$,\nwhere $v \\in \\mathcal{V}$ and $a \\in \\mathcal{F}$ are joined by an edge in $G$ if and only if $v \\in a$. In other words, the vertices of $G$ are the vertices and edges of $H$,\nand the edges of $G$ represent incidences.\n\nTo avoid confusion, we refer to the vertices of a factor graph as \\emph{nodes}. In particular, the nodes in $\\mathcal{V}$ are called \\emph{variable nodes}\nand the nodes in $\\mathcal{F}$ are called \\emph{factor nodes}.\\footnote{In some contexts in the literature,\nfactor nodes may be called \\emph{functional nodes} or \\emph{constraint nodes}.} We define\n$$\nG^r(n,p)\\coloneqq G(H^r(n,p)),\n$$\ni.e.\\ the factor graph of the $r$-uniform binomial random hypergraph $H^r(n,p)$.\n\\end{definition}\n\nNote that if $H$ is an $r$-uniform hypergraph, then the factor nodes of $G(H)$ all have degree~$r$.\nWe will need the following basic fact about the number of factor nodes in $G^r(n,p)$. The proof is a simple application of a Chernoff bound,\nand appears in Appendix~\\ref{app:numberfactornodes} for completeness.\n\n\\begin{proposition}\\label{prop:numberfactornodes}\nLet $d>0$ be a constant and let $p=\\frac{(1+o(1))d}{\\binom{n}{r-1}}$. Then there exists a function $\\omega_0=\\omega_0(n)$ with\n$\\omega_0\\xrightarrow{n\\to\\infty}\\infty$ such that whp\\ the number $m$ of factor nodes\nin $G^r(n,p)$ satisfies\n\\[\nm=\\left(1\\pm \\frac{1}{\\omega_0}\\right)\\frac{dn}{r}.\n\\]\n\\end{proposition}\n\n\n\n\nIt will be more convenient to study the factor graph than the original hypergraph---in order to do this,\nwe need to understand what the structure corresponding to the loose core looks like in the factor graph.\nWe first define the loose core\nof a factor graph and subsequently observe that it does indeed correspond to the loose core of the hypergraph (Definition~\\ref{def:loosecore}).\n\n\n\\begin{definition}[Loose core]\\label{def:loosecorefactorgraph}\nThe \\emph{loose core} $C=C_G$ of a factor graph $G$ is the maximal subgraph of $G$ such that each factor node of $C$\nhas degree $r$ in $C$ and furthermore:\n\\begin{enumerate}[label=\\textnormal{\\textbf{(C\\arabic*')}}]\n\\item\\label{item:isolatednodesfactorgraph} $C$ contains no isolated variable nodes;\n\\item\\label{item:degreeconditionfactorgraphs} Each factor node in $C$ is adjacent to at least two variable nodes of degree at least two in $C$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\\label{prop:loosecorecorrespondence}\nGiven an $r$-uniform hypergraph $H$, the loose core $C_G$ of the factor graph $G=G(H)$ of $H$ is\nidentical to the factor graph of the loose core $C_H$ of $H$.\n\\end{proposition}\n\n\\begin{proof}\nThe condition that each factor node of $C=C_G$ has degree $r$ in $C$\nmeans that $C$ corresponds to a subhypergraph of $H$ (i.e.\\ no edge of\n$H$ has a vertex removed from it without itself being removed).\nSince variable nodes of $G$ correspond to vertices of $H$, condition \\ref{item:isolatednodesfactorgraph} in Definition~\\ref{def:loosecorefactorgraph}\ncorresponds precisely to \\ref{item:loosecorefirstcond} in Definition~\\ref{def:loosecore}.\nFurthermore, condition~\\ref{item:degreeconditionfactorgraphs} in Definition~\\ref{def:loosecorefactorgraph} is directly analogous\nto condition~\\ref{item:loosecoresecondcond} in Definition~\\ref{def:loosecore}.\n\\end{proof}\n\n\nIn view of Proposition~\\ref{prop:loosecorecorrespondence},\nrather than studying the loose core of the hypergraph, we can study the loose core of the corresponding factor graph instead.\nIn fact, even more convenient than this is a slightly different structure.\n\n\\begin{definition}[Reduced core]\\label{def:reducedcore}\nThe \\emph{reduced core} $R=R_G$ of a factor graph $G$ is the maximal subgraph of $G$\nwith no nodes of degree~$1$.\n\\end{definition}\n\nNote that the reduced core is very similar to the $2$-core of $G$---the only difference\nis that we do not delete isolated nodes, so all original nodes are still present. This will be convenient\nsince it means that all nodes have a well-defined degree within the reduced core\n(and have degree zero if and only if they are not in the $2$-core of $G$).\nSimilarly we will want to describe degree distributions within the loose core,\nbut also incorporating nodes which are in fact not contained in the loose core.\nTo avoid confusion and abuse of terminology, we define the \\emph{padded core}.\n\n\\begin{definition}[Padded core]\\label{def:paddedcore}\nThe \\emph{padded core} $P=P_G$ of a factor graph $G$ is the subgraph of $G$ whose\nnodes are the nodes of $G$ and whose edges are the edges of $C_G$.\n\\end{definition}\nIn other words, the padded core\\ $P_G$ is identical to the loose core $C_G$ except that\nall nodes of $G$ are still present. Equivalently, $P_G$ is the maximal subgraph of $G$\nin which each non-isolated factor node has degree~$r$ and is adjacent to at least two\nvariable nodes of degree at least~$2$. The following observation motivates both our definition\nof the padded core\\ and the interpretation of $\\mu_0$ as the proportion of vertices\nof $H^r(n,p)$ which do not lie in the loose core.\n\n\\begin{remark}\\label{rem:paddedcoredegs}\nFor each $j\\in \\mathbb{N}$,\nthe proportion of variable nodes of $G=G^r(n,p)$ which have degree $j$ in the padded core\\ $P_G$ is $\\mu_j$.\n\\end{remark}\n\n\nIt is important to observe that, if we have found the reduced core, it is very easy to reconstruct the padded core,\nand hence also the loose core. \nLet $\\mathcal{F}_R$ be the set of non-isolated factor nodes of the reduced core $R_G$ and let $P^{'}_G$\nbe the factor graph whose nodes are the nodes of $G$ and whose edges are all edges of $G$ incident to $\\mathcal{F}_R$.\nIn other words, $P^{'}_G$ is the factor graph obtained from $R_G$ by adding back in all\nedges of $G$ attached to non-isolated factor nodes of $R_G$.\n\n\\begin{proposition}\\label{prop:reducedtoloose}\n$P^{'}_G = P_G$.\n\\end{proposition}\n\n\\begin{proof}\nLet $R_1$ denote the graph obtained from the padded core $P_G$ of $G$ by removing\nall edges incident to leaves (which must be variable nodes).\nNote that, since any non-isolated factor node in $P_G$ has at least two neighbours of\ndegree at least two, the same is still true in $R_1$.\nFor the sake of intuitive notation, we also denote\n$$\nR_2 \\coloneqq R_G, \\qquad\nP_1 \\coloneqq P_G, \\qquad P_2 \\coloneqq P^{'}_G.\n$$\nOur goal is to show that $P_1=P_2$.\n\nLet us observe that $P_1$ can be obtained from $R_1$ by the same operation\nwith which $P_2$ is obtained from $R_2$, namely by adding in\nedges of $G$ incident to non-isolated factor nodes.\n\nWe next observe that $R_1$ is a subgraph of $G$ with no nodes of degree $1$,\nand therefore $R_1 \\subseteq R_2=R_G$, by the maximality of $R_G$.\nSince the operation constructing $P_i$ from $R_i$ is inclusion-preserving, $R_1 \\subseteq R_2$ implies that $P_1 \\subseteq P_2$. \n\nIt therefore remains to prove that $P_2 \\subseteq P_1$. \nTo this end, we observe that certainly $P_2=P^{'}_G$ is a subgraph of $G$ in which each non-isolated factor node is\nin the $2$-core of $G$, and therefore\nadjacent to at least two variable nodes of degree at least two. Furthermore each non-isolated\nfactor node of $P_2$ has degree $r$ in $C_2$, and since\n$P_1=P_G$ is the maximal subgraph with these two properties, we have $P_2 \\subseteq P_1$, as required.\n\\end{proof}\n\n\nLet us observe one further fact about the transformation from the reduced core $R_G$ to the padded core $P_G=P^{'}_G$:\nalthough this seemed to be dependent on the initial factor graph $G$, in fact the operation\nsimply involves connecting non-isolated factor nodes of $R_G$ to (distinct) isolated variable nodes until each factor node has degree precisely~$r$.\nThis means that given $R_G$, by Proposition~\\ref{prop:reducedtoloose} we can describe $P_G$ (and therefore also the loose core $C_G$)\nentirely, up to the assignment of which\nnodes are leaves. In other words, $R_G$ already contains all of the ``essential'' information\nof both $P_G$ and $C_G$. It will therefore be enough to study $R_G$ rather than $P_G$ or $C_G$, and this turns out to be simpler.\n\n\nNow the main results of this paper are implied by the following theorem about the reduced core $R_G$\nof the factor graph $G= G^r(n,p)$ of the $r$-uniform binomial random hypergraph.\n\n\nFor a non-negative real number $\\lambda$,\nlet us denote by $\\widetilde \\mathrm{Po}(\\lambda)$ the distribution of a random variable $X$ satisfying\n$$\n\\Pr(X=j) = \\begin{cases}\n\\Pr(\\mathrm{Po}(\\lambda)\\le 1) & \\mbox{if }j=0, \\\\\n0 & \\mbox{if }j=1,\\\\\n\\Pr(\\mathrm{Po}(\\lambda)=j) & \\mbox{if } j \\ge 2.\n\\end{cases}\n$$\nIn other words, the $\\widetilde\\mathrm{Po}$ distribution is identical to the $\\mathrm{Po}$ distribution except\nthat values of $1$ are replaced by $0$. We define the $\\widetilde\\mathrm{Bi}$ distribution analogously.\n\n\\begin{theorem}\\label{thm:factor:reducedcoredegs}\nLet $r,d,p,\\rho_*,\\hat \\rho_*$ be as in Section~\\ref{def:basicdefinitions} and let $G= G^r(n,p)$, i.e.\\ the factor graph of $H^r(n,p)$.\nFor each $j\\in\\mathbb{N}$, let $\\varprop{j}$ and $\\factprop{j}$ be the proportion\nof variable nodes and factor nodes of $G$ respectively which have degree $j$ in the reduced core $R_G$ of $G$.\nThen there exists a function $\\eps = \\eps(n) =o(1)$\nsuch that\nwhp\\ for any constant $j \\in \\mathbb{N}$ we have\n$$\n\\varprop{j} = \\Pr(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j)\\pm\\eps\n$$\nand\n$$\n\\factprop{j} = \\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j)\\pm\\eps.\n$$\n\\end{theorem}\nIn other words, in terms of their degrees in $R_G$, variable nodes and factor nodes\nhave degree distributions which are asymptotically those of a $\\widetilde \\mathrm{Po}(d \\hat \\rho_*)$\nand a $\\widetilde \\mathrm{Bi}(r,\\rho_*)$ distribution respectively.\n\nThe proof of this theorem will form the main body of the paper. In Section~\\ref{sec:peeling} we will prove how Theorem~\\ref{thm:factor:reducedcoredegs} follows from two auxiliary statements,\nstating that for some large integer $\\ell$\nthe proportions of variable and factor nodes of degree $j$ in the graph obtained after $\\ell$ rounds of a peeling process\nare approximately the values given in Theorem~\\ref{thm:factor:reducedcoredegs}\n(Lemma~\\ref{lem:mainlemma1}),\nand furthermore not many nodes are deleted after round $\\ell$ (Lemma~\\ref{lem:mainlemma2}). \n\n\\section{Back to hypergraphs: Proofs of main results}\\label{sec:proofofmainresults}\n\\noindent We now show how all of the results of Section~\\ref{sec:mainresults} follow from Theorem~\\ref{thm:factor:reducedcoredegs}. First we deduce our result on the asymptotic degree distribution of vertices in the loose core of $H^r(n,p)$.\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainresultdegrees}]\nWe will apply Theorem~\\ref{thm:factor:reducedcoredegs} to provide us with a function $\\varepsilon$,\nand we will prove Theorem~\\ref{thm:mainresultdegrees} with $\\varepsilon'\\coloneqq\\sqrt{\\varepsilon} + \\frac{1}{\\sqrt{\\omega_0}}$, where\n$\\omega_0=\\omega_0(n)$ is the function given by \nProposition~\\ref{prop:numberfactornodes}.\n\nFor convenience, for any $j\\in\\mathbb{N}$, let us define\n$$\n\\mu_j' \\coloneqq \\begin{cases}\n\\Pr(\\mathrm{Po}(d\\hat \\rho_*)=j) & \\mbox{if } j \\ge 2;\\\\\n\\eta \\cdot \\Pr(\\mathrm{Po}(d\\hat \\rho_*)=j) & \\mbox{if } j=1;\\\\\n\\Pr(\\mathrm{Po}(d\\hat \\rho_*)=0) + (1-\\eta)\\cdot \\Pr(\\mathrm{Po}(d\\hat \\rho_*)=1) & \\mbox{if } j=0.\n\\end{cases}\n$$\nIn other words, $\\mu_j'$ is the ``idealised version'' of $\\mu_j$,\nand our goal is simply to prove that whp, for each $j\\in\\mathbb{N}$ we have $\\mu_j = \\mu_j' \\pm \\varepsilon$.\nSimilarly we also define\n$$\n\\varpropp{j} \\coloneqq \\Pr(\\widetilde{\\mathrm{Po}}(d\\hat \\rho_*)=j),\n$$\nso by Theorem~\\ref{thm:factor:reducedcoredegs} we have $\\varprop{j} = \\varpropp{j} \\pm \\varepsilon$\nwhp for each $j\\in\\mathbb{N}$. The proof of Theorem~\\ref{thm:mainresultdegrees}\nnow simply consists of relating the $\\mu_j$ to the $\\varprop{j}$,\nrelating the $\\mu_j'$ to the $\\varpropp{j}$ and applying Theorem~\\ref{thm:factor:reducedcoredegs}.\nNote that it follows instantly from the definitions\nthat $\\mu_j' = \\varpropp{j}$ for $j\\in\\mathbb{N}_{\\geq 2}$. We will split the proof into three cases.\n\n\\vspace{0.2cm}\n\\noindent \\textbf{Case 1: $j\\ge 2$.}\\\\\nWe start by showing that $\\mu_j=\\varprop{j}$. Observe that by Remark~\\ref{rem:paddedcoredegs},\n$\\mu_j$ is simply\nthe proportion of variable nodes with degree $j$ in the padded core\\ $P_G$ of $G=G^r(n,p)$.\nTheorem~\\ref{thm:factor:reducedcoredegs} tells us the degrees of variable and factor\nnodes in the reduced core $R_G$ of $G$. By Proposition~\\ref{prop:reducedtoloose},\nmoving from $R_G$ to $P_G$ means that \nwe connect all non-isolated factor nodes of $R_G$ to their original neighbours in $G$,\nand any variable nodes which receive additional incident edges in this process have their degrees changed from~$0$ to~$1$.\nIt follows that for $j\\ge 2$, the proportion $\\mu_j$\nof variable nodes in $G$ with degree $j$ in the padded core\\ $P_G$ is precisely equal to $\\varprop{j}$,\nthe proportion of variable nodes in $G$ with degree $j$ in the reduced core $R_G$.\nTherefore \n\\[\n\\mu_j=\\varprop{j}\\numeq{\\text{Th.}~\\ref{thm:factor:reducedcoredegs}}\\varpropp{j} \\pm \\varepsilon=\\mu_j'\\pm \\varepsilon,\n\\]\nand the statement of Theorem~\\ref{thm:mainresultdegrees} is certainly true for $j\\ge 2$ (indeed, we have proved something stronger since $\\varepsilon<\\varepsilon'$).\n\n\\vspace{0.2cm}\n\\noindent \\textbf{Case 2: $j=1$.}\\\\\nTo prove the case $j=1$, we need to check how many isolated variable nodes become leaves when moving from $R_G$ to $P_G$.\nSince by Proposition~\\ref{prop:reducedtoloose} every factor node of $R_G$ with degree $j\\ge 2$ has $r-j$ leaves connected to it, and since whp\nthe number $m$ of factor nodes in total is $m=\\left(1\\pm\\frac{1}{\\omega_0}\\right)\\frac{dn}{r}$ for some growing function\n$\\omega_0\\xrightarrow{n\\to\\infty}\\infty$ by Proposition~\\ref{prop:numberfactornodes}, whp\nthe number of leaves added, which is simply $\\mu_1 n$, satisfies\n\\begin{align}\\label{eq:leaves1}\n\\mu_1 n=\\sum_{j=2}^r (r-j) \\factprop{j} m\n& = \\left(1\\pm\\frac{1}{\\omega_0}\\right)\\frac{dn}{r} \\sum_{j=2}^r (r-j)\\left(\\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j)\\pm \\varepsilon\\right)\\nonumber \\\\\n& = \\frac{dn}{r} \\sum_{j=2}^r (r-j)\\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j) \\pm \\frac{\\varepsilon'n}{2},\n\\end{align} \nwhere the last line follows since $\\frac{1}{\\omega_0},\\varepsilon = o\\left(\\sqrt{\\varepsilon}+\\frac{1}{\\sqrt{\\omega_0}}\\right)=o(\\varepsilon')$.\nThe sum can be estimated using the definition of the $\\widetilde\\mathrm{Bi}$ distribution\nand equations~\\eqref{eq:relatingtherhos} and~\\eqref{eq:relatingrhoswithexpfunction}:\n\\begin{align*}\n&\\sum\\nolimits_{j=2}^r (r-j)\\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j) \\\\\n& \\hspace{2cm} \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:relatingrhoswithexpfunction}}} \\sum\\nolimits_{j=0}^r (r-j)\\Pr(\\mathrm{Bi}(r,\\rho_*)=j) - r(1-\\rho_*)^r - (r-1)r\\rho_*(1-\\rho_*)^{r-1}\\\\\n& \\hspace{2cm} \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:relatingrhoswithexpfunction}}} r (1-\\rho_*)\\left( 1 - (1-\\rho_*)^{r-1} - (r-1)\\rho_*(1-\\rho_*)^{r-2} \\right)\\\\\n& \\hspace{2cm} \\numeq{\\eqref{eq:relatingtherhos},\\eqref{eq:relatingrhoswithexpfunction}} r \\exp(-d\\hat \\rho_*)\\left(\\hat \\rho_* - (r-1)\\rho_*(1-\\rho_*)^{r-2} \\right).\n\\end{align*}\nSubstituting this into~\\eqref{eq:leaves1} gives\n\\begin{align}\\label{eq:leaves2}\n\\mu_1\n& = d\\exp(-d\\hat \\rho_*) \\left(\\hat \\rho_* - (r-1)\\rho_*(1-\\rho_*)^{r-2}\\right) \\pm \\varepsilon'\/2.\n\\end{align}\nOn the other hand, we have\n\\begin{align*}\n\\mu_1' = \\eta \\cdot\\Pr(\\mathrm{Po}(d\\hat \\rho_*)=1) & = \\left(1- \\frac{(r-1)\\rho_* (1-\\rho_*)^{r-2}}{\\hat \\rho_*}\\right) d\\hat \\rho_* \\exp(-d\\hat \\rho_*)\\\\\n& = d\\exp(-d\\hat \\rho_*)\\left(\\hat \\rho_* - (r-1)\\rho_* (1-\\rho_*)^{r-2}\\right),\n\\end{align*}\nwhich combined with~\\eqref{eq:leaves2} tells us that\n\\begin{equation}\\label{eq:deg1proportion}\n\\mu_1 = \\mu_1' \\pm \\varepsilon'\/2,\n\\end{equation}\nwhich is in fact slightly stronger than required.\n\n\\vspace{0.2cm}\n\\noindent \\textbf{Case 3: $j=0$.}\\\\\nFinally to prove the statement for $j=0$, note that\n$\\mu_0 = \\varprop{0} - \\mu_1$ (deterministically).\nFurthermore, we have $\\sum_{j=0}^\\infty \\mu_j' = \\sum_{j=0}^\\infty \\varpropp{j} =1$,\nand we have already observed that $\\mu_j'=\\varpropp{j}$ if $j\\ge 2$,\nand therefore $\\mu_0' + \\mu_1' = \\varpropp{0} + \\varpropp{1}$.\nObserving also that $\\varpropp{1}=0$, we deduce that\n$\\mu_0' = \\varpropp{0} - \\mu_1'$.\nTherefore, applying Theorem~\\ref{thm:factor:reducedcoredegs} (for $j=0$) and~\\eqref{eq:deg1proportion},\nwe obtain\n$$\n\\mu_0 = \\varprop{0}-\\mu_1 = \\varpropp{0} \\pm \\varepsilon - (\\mu_1'\\pm \\varepsilon'\/2) = \\mu_0'\\pm \\varepsilon'\n$$\nas required.\n\\end{proof}\n\n\nWith a little more calculation we can also determine\nthe number of vertices and edges in the loose core, and therefore also prove Theorem~\\ref{thm:mainresultorder}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainresultorder}]\nObserve that the number of vertices in the loose core of $H=H^r(n,p)$ is simply the number of variable nodes\nof $G=G(H)$ which have degree at least one in the padded core\\ of $G$,\nand thus the proportion of such vertices is $1-\\mu_0$ (see Remark~\\ref{rem:paddedcoredegs}). By Theorem~\\ref{thm:mainresultdegrees}, whp\n\\begin{align*}\n1-\\mu_0 & \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:eta},\\eqref{eq:relatingrhoswithexpfunction}}} 1-\\exp(-d\\hat \\rho_*) - d\\hat \\rho_*\\exp(-d\\hat \\rho_*)(1-\\eta)+o(1) \\\\\n& \\numeq{\\eqref{eq:relatingtherhos},\\eqref{eq:eta},\\eqref{eq:relatingrhoswithexpfunction}}\\rho_* - d(1-\\rho_*)(r-1) \\rho_*(1-\\rho_*)^{r-2}+o(1)\\\\\n& \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:eta},\\eqref{eq:relatingrhoswithexpfunction}}}\\rho_* (1 - d(r-1)(1-\\rho_*)^{r-1})+o(1)=\\alpha+o(1),\n\\end{align*}\nprecisely as stated in Theorem~\\ref{thm:mainresultorder}.\n\nThe number of edges in the loose core of $H$ is the number of factor nodes with degree at least $1$ in $R_G$,\nwhich is $(1-\\factprop{0})m$, where recall that $m$ denotes the total number of factor nodes of $G$.\nApplying Theorem~\\ref{thm:factor:reducedcoredegs} to estimate $\\factprop{0}$\nand Proposition~\\ref{prop:numberfactornodes} to estimate $m$,\nwe deduce that\nwhp\\ the number of edges in the loose core is\n\\begin{align*}\n\\left(1-\\factprop{0}\\right)m\n&=\\left(1-(1-\\rho_*)^r-r\\rho_*(1-\\rho_*)^{r-1}\\pm o(1)\\right)\\frac{(1+o(1))dn}{r}\n\\numeq{\\eqref{eq:definitionofeta}}\\left(\\beta+o(1)\\right)n,\n\\end{align*} \nas claimed.\n\\end{proof}\n\nNow we can also prove the bound on the length of the longest loose cycle in Theorem~\\ref{thm:mainresultcycle}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainresultcycle}]\nLet us observe that for any loose cycle in $H=H^r(n,p)$, the edges and the vertices which lie in two edges form\na cycle in the factor graph $G=G(H)$, which must clearly lie within the reduced core $R_G$ of $G$. Thus the\nlength of the loose cycle is bounded both by the number of variable nodes and the number of factor nodes\nwhich are not isolated\nin $R_G$. In other words, the length $L_C$ of the longest loose cycle (deterministically) satisfies\n\\begin{equation}\\label{eq:cyclelengthmin}\nL_C\\le \\min\\left\\{(1-\\varprop{0})n \\; , \\; (1-\\factprop{0}) m\\right\\}.\n\\end{equation}\nBy Proposition~\\ref{prop:numberfactornodes} we have that whp\\ $m=\\left(1+o(1)\\right)\\frac{dn}{r}$.\nObserve also that by Theorem~\\ref{thm:factor:reducedcoredegs}, whp\\ $\\varprop{0}$ is asymptotically\n\\begin{align*}\n\\Pr(\\widetilde{\\mathrm{Po}}(d\\hat \\rho_*)=0) = \\prob(\\mathrm{Po}(d\\hat \\rho_*)\\leq 1) = \\exp(-d\\hat \\rho_*)(1+d\\hat \\rho_*)=1-\\gamma,\n\\end{align*}\nwhile whp $\\factprop{0}$ is asymptotically\n\\begin{align*}\n\\Pr(\\widetilde{\\mathrm{Bi}}(r,\\rho_*)=0) = \\prob(\\mathrm{Bi}(r,\\rho_*)\\leq 1) = (1-\\rho_*)^r + r\\rho_*(1-\\rho_*)^{r-1}=1-\\frac{\\beta r}{d}.\n\\end{align*}\nSubstituting these values into~\\eqref{eq:cyclelengthmin} gives the bound in Theorem~\\ref{thm:mainresultcycle}.\n\\end{proof}\n\nOur next goal is to prove the remaining results of Section~\\ref{sec:mainresults},\nfor which we will need to relate $L_P$ and $L_C$. To do this, we use a standard sprinkling argument.\n\n\\begin{lemma}\\label{lem:standardsprinkling}\nLet $\\omega = \\omega(n)$ be any function and\n$p_1=p_1(n)$ and $p_2=p_2(n)$ be any probabilities satisfying\n\\begin{enumerate}\n\\item $p_1 \\le \\left(1+\\frac{1}{\\omega}\\right)p_1 \\le p_2;$\n\\item $p_1 n^r \/\\omega \\to \\infty$.\n\\end{enumerate}\nSuppose that $H_1 \\sim H^r(n,p_1)$ and $H_2 \\sim H^r(n,p_2)$ are coupled in such a way that $H_1 \\subset H_2$.\nFor $i=1,2$, let $L_P^{(i)},L_C^{(i)}$ denote the length of the longest loose path and loose cycle, respectively, in $H_i$. Then whp\\ \\[L_C^{(2)} \\ge L_P^{(1)}+o(n).\\]\n\\end{lemma}\nWe defer the proof of this lemma to Appendix~\\ref{app:sprinkling}.\nThe following slightly different form will be a little more convenient to apply.\nWe omit the proof, which is elementary given Lemma~\\ref{lem:standardsprinkling}. \\pagebreak\n\\begin{corollary}\\label{cor:sprinkling}\nGiven the setup of Lemma~\\ref{lem:standardsprinkling}, the following hold.\n\\begin{enumerate}\n\\item If there exists a constant $\\zeta_1$ such that whp\\ $L_P^{(1)} \\ge (\\zeta_1 +o(1))n$, then whp\\ $L_C^{(2)} \\ge (\\zeta_1 +o(1))n$.\n\\item If there exists a constant $\\zeta_2$ such that whp\\ $L_C^{(2)} \\le (\\zeta_2 +o(1))n$, then whp\\ $L_P^{(1)} \\le (\\zeta_2 +o(1))n$.\n\\end{enumerate}\n\\end{corollary}\n\n\n\nWe also need a further technical result which states that the parameters $\\beta,\\gamma$,\nwith which we bound $L_C$ in Theorem~\\ref{thm:mainresultcycle}, are continuous in $p$\n(except at the threshold $p=d^*\/\\binom{n-1}{r-1}$).\nLet $r,d,p$ be as in Section~\\ref{def:basicdefinitions}. \nLet $p'=(1+1\/\\omega)p$ for a function $\\omega=\\omega(n)\\to\\infty$ but $\\omega=o(\\log n)$.\nThe following lemma shows that if we replace $p$ by $p'$,\nthe parameters $\\beta,\\gamma$ remain essentially the same. The (technical) proof can be found in Appendix~\\ref{app:analysisfixedpoint}.\n\\begin{lemma}\\label{lem:continuity}\nLet $\\beta,\\gamma$ be defined as in~\\eqref{eq:definitionofeta} and~\\eqref{eq:definitionofgamma},\nand let $\\beta',\\gamma'$ be defined similarly but with $p'$ in place of $p$. If $d\\neq d^*,$ then \n\\[\\min\\{\\beta',\\gamma'\\}=\\min\\{\\beta,\\gamma\\}+o(1).\\]\n\\end{lemma}\n\n\n\nWe can now bound the length of the longest loose path in $H^r(n,p)$.\n\\begin{proof}[Proof of Corollary~\\ref{cor:upperboundlongestpath}]\nLet us set $\\omega = 1\/(\\log n)$, set $p_1=p$ and set $p_2 = \\left(1+\\frac{1}{\\omega}\\right)p_1$.\nIt is easy to check that these parameters satisfy the assumptions of Lemma~\\ref{lem:standardsprinkling},\nand therefore also of Corollary~\\ref{cor:sprinkling}.\nTheorem~\\ref{thm:mainresultcycle} applied to $H_2 \\sim H^r(n,p_2)$ implies that whp\\ $L_C^{(2)} \\le (\\min\\{\\beta_2,\\gamma_2\\} + o(1))n$,\nwhere $\\beta_2,\\gamma_2$ are defined analogously to $\\beta,\\gamma$, but with $p_2 = (1+1\/\\omega)p$ in place of $p$.\nFurthermore, Lemma~\\ref{lem:continuity} implies that $\\min\\{\\beta_2,\\gamma_2\\} = \\min\\{\\beta,\\gamma\\}+o(1)$,\nso we deduce that whp\\ $L_C^{(2)} \\le (\\min\\{\\beta,\\gamma\\} + o(1))n$.\nFinally, Corollary~\\ref{cor:sprinkling} then implies that whp\\ $L_P=L_P^{(1)} \\le (\\min\\{\\beta,\\gamma\\} + o(1))n$,\nas required.\n\\end{proof}\n\nBy applying Corollary~\\ref{cor:upperboundlongestpath} shortly beyond the phase transition\nthreshold, we are able to prove Corollary~\\ref{cor:shortlyafterphasetransition}.\n\\begin{proof}[Proof of Corollary~\\ref{cor:shortlyafterphasetransition}]\nSince $\\min\\{\\beta,\\gamma\\}\\leq \\gamma$ it suffices to show that \n\\[\\gamma\\leq \\frac{2\\varepsilon^2}{(r-1)^2}+O(\\varepsilon^3).\\] (In fact a similar computation for $\\beta$ gives exactly the same result.)\nBy definition\n\\begin{align}\n \\gamma &\\numeq{\\eqref{eq:definitionofgamma}} 1-\\exp(-d\\hat \\rho_*)-d\\hat \\rho_* \\exp(-d\\hat \\rho_*) \\nonumber \\\\\n & = 1-\\left(1-d\\hat \\rho_*+\\frac{d^2\\hat \\rho_*^2}{2}+O\\left(\\hat \\rho_*^3\\right)\\right)-d\\hat \\rho_*\\left(1-d\\hat \\rho_*+O\\left(\\hat \\rho_*^2\\right)\\right) \\nonumber \\\\\n &= \\frac{d^2\\hat \\rho_*^2}{2}+O\\left(\\hat \\rho_*^3\\right) \\label{eq:gammadefapprox}\n\\end{align}\nRecall from~\\eqref{eq:relatingtherhos} that $\\hat \\rho_*$ was defined as a function of $\\rho_*$,\nwhich itself was defined as the largest solution of the fixed-point equation~\\eqref{eq:fixedpointequation}.\nWe therefore need to estimate $\\rho_*$.\nFrom~\\eqref{eq:fixedpointequation} we obtain\n\\begin{align*}\n \\rho &= \\frac{-d(r-1)+1}{(-\\frac{1}{2}-\\frac{d}{2}(r-1)(r-2))}+O\\left(\\rho^2\\right).\\label{eq:rhoapproximation}\n\\end{align*}\n\n\\noindent Substituting $d=\\frac{1+\\varepsilon}{r-1}$ gives\n\\begin{equation*}\n \\rho=\\frac{2\\varepsilon}{1+(1+\\varepsilon)(r-2)}+O\\left(\\rho^2\\right)=\\frac{2\\varepsilon}{r-1+O(\\varepsilon)}+O\\left(\\rho^2\\right)=\\frac{2\\varepsilon}{r-1}+O\\left(\\rho^2\\right).\n\\end{equation*}\nIn particular this implies that there exists a solution $\\rho=\\frac{2\\varepsilon}{r-1}+O(\\varepsilon^2)$\nof the fixed point equation~\\eqref{eq:fixedpointequation},\nand by Claim~\\ref{claim:behaviouroffixedpointsol} this is the unique positive solution\nand therefore\n$\\rho_*=\\frac{2\\varepsilon}{r-1}+O(\\varepsilon^2)$.\nSubstituting this into~\\eqref{eq:relatingtherhos} we obtain\n\\begin{equation*}\n\\hat \\rho_*=1-\\left(1-\\frac{2\\varepsilon}{r-1}+O(\\varepsilon^2)\\right)^{r-1}=2\\varepsilon+O(\\varepsilon^2).\n\\end{equation*}\nSubstituting this into~\\eqref{eq:gammadefapprox}, we obtain\n\\begin{align*}\n \\gamma &= \\frac{2\\varepsilon^2}{(r-1)^2}+O\\left(\\varepsilon^3\\right).\\qedhere\n\\end{align*}\n\\end{proof}\n\nIn order to prove Theorem~\\ref{thm:bestknownresultcycles},\nwe also need a lower bound on $L_C$.\nWe will use a result of~\\cite{cooley2020longest},\nwhich provides a lower bound on $L_P$ together with Lemma~\\ref{lem:standardsprinkling} to relate $L_P$ and $L_C$.\nMore precisely, one special case (the supercritical regime for $j=1$) of~\\cite[Theorem~4]{cooley2020longest} can be\nreformulated (in a slightly weakened but much simplified way) as follows.\n\n\n\n\\begin{theorem}[\\!\\!\\cite{cooley2020longest}]\\label{thm:pathsresult}\nLet $L_P$ denote the length of the longest loose path in $H^r(n,p)$. \nFor all $r\\in \\mathbb{N}_{\\ge 3}$ there exists $\\varepsilon_0 \\in (0,1]$ such that for any function\n$\\varepsilon=\\varepsilon(n)<\\varepsilon_0$ which satisfies\n$\\varepsilon^5 n \\xrightarrow{n\\to \\infty} \\infty$, setting $\\delta=\\varepsilon\/\\sqrt{\\varepsilon_0}$\nthe following holds. \nIf $p=\\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$,\nthen whp\n \\[\n (1 - \\delta)\\frac{\\varepsilon^2 n}{4(r-1)^2} \\leq L_P \\leq (1 + \\delta)\\frac{2 \\varepsilon n}{(r-1)^2}.\n \\]\n\\end{theorem}\n\nNote that Theorem~\\ref{thm:pathsresult} allows for a wider range of $\\varepsilon$ than we consider in this paper,\nin particular allowing $\\varepsilon$ to tend to zero sufficiently slowly. However, there is a $\\Theta(\\varepsilon)$ gap\nbetween the upper and lower bounds. Theorem~\\ref{thm:bestknownresultcycles} improves the upper bound\nand thus narrows the gap to just a constant factor.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bestknownresultcycles}]\nThe second and third inequalities are simply the statement of Corollary~\\ref{cor:upperboundlongestpath},\nso it remains to show that whp\n\\[\n\\left(\\frac{\\varepsilon^2}{4(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n\\leq L_C.\n\\]\nNote that we may assume that $\\varepsilon<\\varepsilon_0$, where $\\varepsilon_0$ is the parameter from\nTheorem~\\ref{thm:pathsresult},\nsince otherwise the $O(\\varepsilon^3)$ error term may in fact be the dominant term, and the result is trivial.\n\nLet us set $p_2 = p$ and $p_1 = \\left(1-\\frac{1}{\\log n}\\right)p$. It is easy to check that these parameters\nsatsify the assumptions of Lemma~\\ref{lem:standardsprinkling}, and therefore also of Corollary~\\ref{cor:sprinkling}.\n\nIt is also clear that $p_1 = \\frac{1+\\varepsilon_1}{(r-1)\\binom{n-1}{r-1}}$, where $\\varepsilon_1 = \\varepsilon-\\frac{1}{\\log n}-\\frac{\\varepsilon}{\\log n}=\\varepsilon + O(\\varepsilon^2)$,\nand\ntherefore the lower bound in Theorem~\\ref{thm:pathsresult} (together with the observation that $\\varepsilon_1\/\\sqrt{\\varepsilon_0} = O(\\varepsilon_1)$)\nstates that whp\\ \n$$\nL_P^{(1)} \\ge \\left(\\frac{\\varepsilon_1^2}{4(r-1)^2} + O(\\varepsilon_1^3)\\right)\\cdot n = \\left(\\frac{\\varepsilon^2}{4(r-1)^2} + O(\\varepsilon^3)\\right)\\cdot n,\n$$\nand an application of Corollary~\\ref{cor:sprinkling} completes the proof.\n\\end{proof}\n\n\n\n\n\\section{Peeling process}\\label{sec:peeling}\nRecall that for a given hypergraph $H$ the reduced core of the factor graph $G=G(H)$\nis defined as the maximum subgraph with no nodes of degree one, which is similar to the $2$-core of $G$\nexcept that isolated nodes are not deleted.\nThere is a simple peeling process to obtain the $2$-core of $G$ which is a standard procedure and has been used and analysed extensively in the literature.\nWe will consider the obvious adaptation of this process which obtains the reduced core rather than the $2$-core.\n\n\\begin{definition}[Peeling Process]\\label{def:peeling}\nIn every round we check whether the factor graph has any nodes of degree one\nand delete edges incident to such nodes. More precisely, we recursively define a sequence of graphs $(G_i)_{i\\in\\mathbb{N}}$\nwhere $G_0$ is the input graph and for $i\\in\\mathbb{N}_{\\geq 1}$, $G_i$ is the graph obtained from $G_{i-1}$\nby removing all edges incident to nodes of degree one. We say that we \\emph{disable} a node\nif we delete its incident edges.\n\n\n\\end{definition} \n\\noindent Note that there exists an $i_0$ such that $G_{i_0}=G_{i_0+k}=R_{G_0}$ for any $k\\in\\mathbb{N}$. \\pagebreak\n\nWe recall the definition of $\\varprop{j}$ and $\\factprop{j}$ in Theorem~\\ref{thm:factor:reducedcoredegs} and observe that \\[\\varprop{j}\\coloneqq\\lim\\limits_{\\ell\\to\\infty}\\varpropl{j}{\\ell} \\text{ \\ and \\ } \\factprop{j}\\coloneqq\\lim\\limits_{\\ell\\to\\infty}\\factpropl{j}{\\ell},\\] \nwhere $\\varpropl{j}{\\ell}, \\factpropl{j}{\\ell}$ are the proportions of variable nodes and factor nodes respectively which have degree $j$ in $G_{\\ell}$ for $\\ell\\in\\mathbb{N}$. These limits exist since both $\\left(\\varpropl{j}{\\ell}\\right)_{\\ell}$ and $\\left(\\factpropl{j}{\\ell}\\right)_{\\ell}$ remain constant after a finite number of steps.\n\n\nWe will prove Theorem~\\ref{thm:factor:reducedcoredegs} with the help of two lemmas. The first describes the asymptotic distribution of $\\varpropl{j}{\\ell}$ and $\\factpropl{j}{\\ell}$ for large $\\ell$.\n\\begin{lemma}\\label{lem:mainlemma1}\nLet $r,d,\\rho_*,\\hat \\rho_*$ be as in Section~\\ref{def:basicdefinitions}.\nThere exist an integer $\\ell= \\ell(n) \\in \\mathbb{N}$ and a real number $\\varepsilon_1=\\varepsilon_1(n)=o(1)$\nsuch that whp, for any constant $j\\in\\mathbb{N}$ \n\\[\n\\varpropl{j}{\\ell}=\\prob(\\widetilde \\mathrm{Po}(d\\hat \\rho_*)=j)\\pm\\varepsilon_1\n\\]\nand\n\\[\n\\factpropl{j}{\\ell}=\\prob(\\widetilde \\mathrm{Bi}(r,\\rho_*)=j)\\pm\\varepsilon_1.\n\\]\n\\end{lemma}\n\nThe second lemma states that $\\varpropl{j}{\\ell}$ and $\\factpropl{j}{\\ell}$ approximate $\\varprop{j}$ and $\\factprop{j}$, respectively.\n\\begin{lemma}\\label{lem:mainlemma2}\nLet $r,d$ be as in Section~\\ref{def:basicdefinitions}.\nFor each $j\\in \\mathbb{N}$,\nlet $\\varprop{j},\\factprop{j}$ be as defined in Theorem~\\ref{thm:factor:reducedcoredegs},\nlet $\\ell,\\varepsilon_1$ be as in Lemma~\\ref{lem:mainlemma1} and set $\\varepsilon_2\\coloneqq\\sqrt{\\varepsilon_1}.$ Then whp\\ the peeling process will disable at most $\\varepsilon_2 n$ nodes after round $\\ell$. In particular whp, for any constant $j\\in\\mathbb{N}$\n\\[\n\\varprop{j}=\\varpropl{j}{\\ell}\\pm\\varepsilon_2\n\\]\nand\n\\[\n\\factprop{j}=\\factpropl{j}{\\ell}\\pm \\frac{2\\varepsilon_2r}{d} .\n\\]\n\\end{lemma}\nBefore proving these two lemmas, we show how together they imply our main result. \n\\begin{proof}[Proof of Theorem~\\ref{thm:factor:reducedcoredegs}]\nLet $\\ell,\\varepsilon_1,\\varepsilon_2$ be as in Lemmas~\\ref{lem:mainlemma1} and~\\ref{lem:mainlemma2}.\nApplying these two lemmas, whp\\ we have\n$$\n\\varprop{j} \\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma2}}}{=} \\varpropl{j}{\\ell} \\pm \\varepsilon_2\n\\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma1}}}{=} \\Pr(\\widetilde \\mathrm{Po}(d\\hat \\rho_*)=j) \\pm (\\varepsilon_1 + \\varepsilon_2).\n$$\nSimilarly, whp\\ we have\n$$\n\\factprop{j} \\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma2}}}{=} \\factpropl{j}{\\ell} \\pm \\frac{2\\varepsilon_2 r}{d}\n\\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma1}}}{=} \\prob(\\widetilde \\mathrm{Bi}(r,\\rho_*\\, )=j)\\pm \\left(\\varepsilon_1 + \\frac{2\\varepsilon_2 r}{d}\\right).\n$$\nThe statement of Theorem~\\ref{thm:factor:reducedcoredegs} follows by setting $\\varepsilon = \\varepsilon_1+\\varepsilon_2\\max\\{1,2r\/d\\}$.\n\\end{proof}\n\n\n\n\n\n\n\\section{\\coredetector \\ Algorithm: Proof of Lemma~\\ref{lem:mainlemma1}}\\label{sec:algorithm}\n\\subsection{Main algorithm}\nIn this section we will introduce the \\coredetector \\ algorithm,\nwhich is related to the peeling process. To do so, we need to define some notation---this\nnotation could apply to any graph, but since we will need it specifically for factor graphs, we introduce it in this\n(slightly restrictive) setting for clarity.\n\\begin{definition}\\label{def:algorithmdefs}\nLet $G$ be a factor graph with variable node set $\\mathcal{V}$ and factor node set $\\mathcal{F}$.\nWe denote by $d_G(u,v)$ the distance between two nodes $u,v\\in \\mathcal{V} \\cup \\mathcal{F}$, i.e.\\ the number of edges in a shortest path between them.\nFor each $\\ell\\in\\mathbb{N}$ and each $w\\in \\mathcal{V}\\cup\\mathcal{F}$, we define\n\\[\nD_{\\ell}(w)\\coloneqq\\{u\\in\\mathcal{V}\\cup\\mathcal{F}:d_{G}(w,u)=\\ell\\}\n\\]\nand\n\\[\nd_{\\ell}(w)\\coloneqq|D_{\\ell}(w)|.\n\\]\n\\pagebreak \n\n\\noindent Let \n\\[\nD_{\\leq\\ell}(w)=\\bigcup_{i=0}^\\ell D_i(w)\n\\]\nand\n\\[N_{\\leq\\ell}(w)\\coloneqq G[D_{\\leq\\ell}(w)],\\] i.e.\\ the subgraph of $G$ induced on $D_{\\leq\\ell}(w)$.\n\\end{definition}\n\nWe consider a procedure called \\coredetector .\nGiven a factor graph $G$ on node set $\\mathcal{V} \\cup \\mathcal{F}$\nand a node $w \\in \\mathcal{V} \\cup \\mathcal{F}$,\nwe consider the factor graph as being rooted at $w$. In particular, neighbours of a node $v$ which are at distance $d_G(v,w)+1$ from $w$\nare called \\emph{children} of $v$.\nStarting at distance $\\ell \\in \\mathbb{N}$ and moving up towards the root $w$, we recursively delete any node with no (remaining) children;\nAlgorithm~\\ref{algorithm:coredetector 2} gives a formal description of this procedure.\nWe will denote by $D^*_{\\ell-i}(w)$ the set of nodes in $D_{\\ell-i}(w)$ which survive round $i$ and let $d^*_{i}(w)\\coloneqq|D^*_{i}(w)|$. \n\\begin{algorithm}\n\t\\DontPrintSemicolon\n\t\\KwIn{Integer $\\ell\\in\\mathbb{N}$, node $w\\in\\mathcal{V}\\cup\\mathcal{F}$, factor graph $N_{\\leq\\ell+1}(w)$}\n\t\\KwOut{$d^*_1(w)$}\n $D^*_{\\ell+1}(w)=D_{\\ell+1}(w)$\\;\n\t\\For{$1\\leq i\\leq \\ell$}{\n $D^*_{\\ell-i+1}(w)\\gets D_{\\ell-i+1}(w)\\setminus\\ \\Big\\{v:N(v)\\capD^*_{\\ell-i+2}(w)=\\emptyset\\Big\\}$\\;\n\t$d^*_{\\ell-i+1}(w)\\gets|D^*_{\\ell-i+1}(w)|$}\n\t\t\\caption{\\textnormal{\\texttt{CoreConstruct}} }\n\t\\label{algorithm:coredetector 2}\n\\end{algorithm}\n\nIt is rather difficult to analyse the peeling process directly and it turns out that \\coredetector \\ is easier to analyse\nwhile also being closely related.\n\\coredetector \\ is intended to model the effect of the peeling process on the degree of $w$ after $\\ell$ steps\n(although note that \\coredetector \\ does delete nodes rather than merely disabling them).\nNote, however, that it does not mirror the peeling process precisely; some nodes may be disabled in the peeling process much\nearlier than they are deleted in \\coredetector ,\nand some nodes may be deleted in \\coredetector \\ even\nthough they are actually in the reduced core, and are therefore never disabled in the peeling process.\nNevertheless, we obtain the following important relation.\nRecall that $G_{\\ell}$ is the graph obtained after the $\\ell$-th round of the peeling process (see Definition~\\ref{def:peeling})\nand that $d_{G_\\ell}(w)$ is the degree of the node $w$ in $G_\\ell$.\n\n\\begin{lemma}\\label{lem:algvspeeling}\nLet $\\ell\\in\\mathbb{N}_{\\geq 1}$ and $w\\in\\mathcal{V}\\cup\\mathcal{F}$. If there are no cycles in $N_{\\leq\\ell+1}(w)$, then\nthe output $d^*_1(w)$ of \\coredetector \\ \nwith input $\\ell,w$ and $N_{\\le \\ell+1}(w)$ satisfies\n\\[d_{G_{\\ell}}(w) \\begin{cases}\n=d^*_1(w) & \\text{if }d^*_1(w)\\neq 1, \\\\\n\\leq d^*_1(w) & \\text{if }d^*_1(w)=1.\n\\end{cases}\n\\]\n\\end{lemma}\n\\begin{proof}\nFor an upper bound,\nwe will show that if a given node $v\\in\\mathcal{V}\\cup\\mathcal{F}$ is deleted in round $i$ of \\coredetector , \nit must have been disabled at some round $i'\\leq i$ in the peeling process for the reduced core.\nIn particular, by setting $i=\\ell$ we immediately obtain\n$d_{G_{\\ell}}(w)\\leqd^*_1(w).$\nWe prove the statement by induction on $i$.\n\nFor $i=1$, if a node $v$ is deleted in round one of \\coredetector , then $v$ had no children,\nand therefore it has only one neighbour in $G=G_0$ (its parent in $N_{\\le \\ell+1}(w)$).\nThus $v$ will be disabled in round one\nof the peeling process.\nNow suppose $v$ is deleted in round $i\\ge 2$ of \\textnormal{\\texttt{CoreConstruct}}, which must mean that all its children\n(if it had any) are deleted in step $i-1$ of \\textnormal{\\texttt{CoreConstruct}}.\nBy the induction hypothesis, all its children are disabled by step at most $i-1$ of the peeling process\nand so have degree $0$ in $G_{i-1}$.\nTherefore $v$ itself has degree at most one in $G_{i-1}$ (from its parent)\nand so will be disabled in round $i$ of the peeling process if it has not been disabled already.\n\n\nIt remains to prove that \n$d_{G_{\\ell}}(w)\\geqd^*_1(w)$\nif $d^*_1(w)\\geq 2$. Let $j\\coloneqqd^*_1(w)\\ge 2$ be the number of children of the root $w$ which survive \\coredetector .\nEach such child must have a descendant in $D_{\\ell+1}(w)$, otherwise it would not survive \\coredetector .\nThus we have $j$ paths of length $\\ell+1$ which all meet at $w$, but are otherwise disjoint (since $N_{\\le \\ell+1}(w)$ contains no cycles).\nBy induction on $i$, we deduce that after $i$ rounds of the peeling process,\nthere are $j$ paths of length $\\ell+1-i$ which meet only in $w$,\nand in particular after $\\ell$ rounds of the peeling process, $d_{G_\\ell}(w) \\ge j$, as required.\n\\end{proof}\n\nFrom now on for the rest of this section, we will always have $G=G^r(n,p)$.\nObserve that if $d_{G_\\ell}(w)\\in \\{0,1\\}$, then $d_{R_G}(w)=0$. However, if $\\ell$ is sufficiently large\nwe can even say that\nin $G_\\ell$ the degree will almost always be $0$.\n\n\\begin{proposition}\\label{prop:1rare}\nFor any integer-valued function $\\ell=\\ell(n) \\xrightarrow{n\\to \\infty} \\infty$ and node $w$ we have\n$\\Pr\\left(d_{G_\\ell}(w)=1\\right) = o(1)$.\n\\end{proposition}\n\n\\begin{proof}\nLet us first assume that $w$ is a variable node.\nFor any integer $i \\ge 1$, let $\\mathcal{V}_i$ and $\\mathcal{F}_i$ be the set of variable nodes and factor nodes respectively which are disabled in round~$i$\nof the peeling process.\nIt is an elementary fact about the peeling process that for any integer $i \\ge 2$\nwe have\n$|\\mathcal{V}_i| \\le |\\mathcal{F}_{i-1}|$ and $|\\mathcal{F}_i| \\le |\\mathcal{V}_{i-1}|$ (deterministically),\nfrom which it follows that $|\\mathcal{V}_i|\\le |\\mathcal{V}_{i-2}|$ for $i \\ge 3$.\nTherefore we have\n$$\n|\\mathcal{V}_\\ell| \\le |\\mathcal{V}_{\\ell-2}| \\le \\ldots \\le |\\mathcal{V}_{\\ell -2\\lfloor \\frac{\\ell-1}{2}\\rfloor}|.\n$$\nFurthermore, the $\\mathcal{V}_i$ are all disjoint, and so we have\n$$|\\mathcal{V}_\\ell| \\le \\frac{n}{1+\\lfloor \\frac{\\ell-1}{2}\\rfloor} = O(n\\ell^{-1}) = o(n)$$\ndeterministically. Therefore for large enough $n$ we have\n$|\\mathcal{V}_\\ell| \\in K \\coloneqq \\left[n\/\\sqrt{\\ell}\\right]_0$ and so by symmetry we have\n$$\n\\Pr\\left(w \\in \\mathcal{V}_\\ell\\right) = \\sum_{k \\in K} \\left( \\Pr\\left(|\\mathcal{V}_\\ell|=k\\right) \\cdot \\frac{k}{n} \\right) \\le \\frac{1}{\\sqrt{\\ell}} \\cdot \\sum_{k \\in K} \\Pr\\left(|\\mathcal{V}_\\ell|=k\\right) = o(1),\n$$\nas required.\nThe proof when $w$ is a factor node is essentially identical.\n\\end{proof}\n\n\n\\subsection{Analysis of \\coredetector }\n\nWe proceed with the analysis of \\coredetector \\ and we will choose the parity of $\\ell$ such that $D_{\\ell+1}(w)$ consists of variable nodes,\ni.e.\\ if $w\\in \\mathcal{V}$, then we will choose $\\ell$ odd, and if $w \\in \\mathcal{F}$ we will choose $\\ell$ to be even.\nThis convention is merely for technical convenience since it ensures that we\nknow which type of nodes are being considered in round $i$ of \\textnormal{\\texttt{CoreConstruct}}\\ and thus avoid\na case distinction.\n\nLet $w\\in\\mathcal{V}\\cup\\mathcal{F}$ and $\\ell\\in\\mathbb{N}_{\\geq 1}$ be given. We say the event $E_w(\\ell)$ holds if \\emph{neither} of the following two events occur. \n\\begin{enumerate}[label=\\textnormal{\\textbf{(E\\arabic*)}}]\n \\item $|D_{\\leq \\ell+1}(w)|\\geq(\\log n)^2$;\n \\item $D_{\\leq \\ell+1}(w)$ contains a node which lies on a cycle of length at most $2\\ell$.\n\\end{enumerate}\nWe will later condition on the event $E_{w}(\\ell)$ holding, and therefore need to know that it is very likely.\n\n\\begin{lemma}\\label{lem:eventE}\nFor any function $\\ell=o(\\log\\log n)$ and node $w \\in \\mathcal{V} \\cup \\mathcal{F}$,\n$$\n\\Pr\\left(E_{w}(\\ell)\\right) \\ge 1-\\exp\\left(-\\Theta\\left(\\sqrt{\\log n}\\right)\\right).\n$$\nFurthermore, whp\\ all but $o(n)$ nodes $w\\in\\mathcal{V}\\cup\\mathcal{F}$ satisfy $E_{w}(\\ell)$.\n\\end{lemma}\n\nThe (standard) proof appears in Appendix~\\ref{app:eventE}.\n\nNow let us define\n$$\n\\tilde d_1(w):=\n\\begin{cases}\nd^*_1(w) & \\mbox{if }d^*_1(w) \\neq 1\\\\\n0 & \\mbox{if }d^*_1(w) =1.\n\\end{cases}\n$$\nIn other words, $\\tilde d_1(w)$ is identical to $d^*_1(w)$ except that values of $1$ are replaced by $0$\n(similar to the $\\widetilde\\mathrm{Po}$ and $\\widetilde\\mathrm{Bi}$ distributions compared to the $\\mathrm{Po}$ and $\\mathrm{Bi}$ distributions).\nWe can combine Lemmas~\\ref{lem:eventE} and~\\ref{lem:algvspeeling} and Proposition~\\ref{prop:1rare}\nto obtain the following.\n\n\\begin{corollary}\\label{cor:Gldegalmosteq}\nFor any integer-valued function $\\ell=\\ell(n) \\xrightarrow{n\\to \\infty} \\infty$ which also satisfies $\\ell = o(\\log \\log n)$ and any node $w$ we have\n$$\n\\Pr\\left(d_{G_\\ell}(w) \\neq \\tilde d_1(w)\\right) = o(1).\n$$\n\\end{corollary}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:algvspeeling}, the only cases in which $d_{G_\\ell}(w)$ and $\\tilde d_1(w)$ can differ are if $E_{w}(\\ell)$ does not hold or if $d_{G_\\ell}(w)=1$.\nThus by applying Lemma~\\ref{lem:eventE} and Proposition~\\ref{prop:1rare}, we obtain\n\\begin{align*}\n\\Pr\\left(d_{G_\\ell}(w) \\neq \\tilde d_1(w)\\right) & \\le \\Pr\\left(\\bar{E}_w(\\ell)\\right) + \\Pr\\left(d_{G_\\ell}(w) =1\\right)\\\\\n& \\le \\exp\\left(-\\Theta\\left(\\sqrt{\\log n}\\right)\\right) + o(1) = o(1).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\nWe next describe the survival probabilities of internal (i.e.\\ non-root) variable and factor nodes in each round of \\textnormal{\\texttt{CoreConstruct}}.\nRecall that for any $i\\in[\\ell]$\nthe set $D^*_{\\ell+1-i}(w)$ consists of nodes within $D_{\\ell+1-i}(w)$ which survive the $i$-th round of \\coredetector . We define the recursions\n\\begin{align*}\n \\intvarsurv{0} &= 1,\\\\\n \\intfactsurv{t} &= \\prob(\\mathrm{Bi}(r-1,\\intvarsurv{t-1})\\geq 1),\\\\\n \\intvarsurv{t} &= \\prob(\\mathrm{Po}(d\\intfactsurv{t})\\geq 1).\n\\end{align*}\n\\begin{lemma}\\label{lem:survivalprobs}\nLet $w\\in\\mathcal{V}\\cup\\mathcal{F}$ and $\\ell$ be odd if $w\\in\\mathcal{V}$ and even if $w\\in\\mathcal{F}$. Let $t\\in\\mathbb{N}$ with $\\ 0\\leq t\\leq \\frac{\\ell+1 }{2}$ be given.\nConditioned on the event $E_{w}(\\ell)$:\n\\begin{enumerate}\n\\item\nFor each $u\\in D_{\\ell+1-2t}(w)$ independently of each other we have\n\\[\n\\prob[u\\inD^*_{\\ell+1-2t}(w)]=\\intvarsurv{t}+o(1);\n\\]\n\\item\nFor each $a\\in D_{\\ell-2t}(w)$ independently of each other we have\n\\[\n\\prob[a\\inD^*_{\\ell-2t}(w)]=\\intfactsurv{t+1}+o(1).\n\\]\n\\end{enumerate}\nIn particular:\n\\begin{enumerate}[(i)]\n\\item If $w \\in \\mathcal{F}$ and $t_1=\\ell\/2$, then for each $u \\in D_1(w)$ independently of each other,\n$$\n \\prob[u\\inD^*_{1}(w)]=\\intvarsurv{t_1}+o(1);\n$$\n\n\\item\\label{eq:probfactnodesurvives} If $w \\in \\mathcal{V}$ and $t_2=(\\ell+1)\/2$, then for each $a\\in D_1(w)$ independently of each other,\n$$\n \\prob[a\\inD^*_{1}(w)]=\\intfactsurv{t_2}+o(1).\n$$\n\\end{enumerate}\n\\end{lemma}\n\nTo prove this lemma, \nwe will need the asymptotic degree distribution of a variable node in $N_{\\leq\\ell}(w)$, which is a standard result.\n\\begin{proposition}\\label{prop:distributionofnochildren}\n Let $w\\in\\mathcal{V}\\cup \\mathcal{F}$ and an integer $\\ell=o(\\log\\log n)$ be given. Conditioned on the event $E_{w}(\\ell)$,\n for each $u \\in D_{\\le \\ell}(w)\\cap \\mathcal{V}$ independently, the number of children of $u$ in $N_{\\leq\\ell+1}(w)$\n is asymptotically distributed as $\\mathrm{Po}(d)$.\n\\end{proposition}\n\nWe defer the proof to Appendix~\\ref{app:offspringdist}. We can now prove Lemma~\\ref{lem:survivalprobs}.\n \n\\begin{proof}[Proof of Lemma~\\ref{lem:survivalprobs}]\nWe will prove both statements (1) and (2) by a common induction on $t$.\nFor $t=0$ the statements are clear since we have $\\intvarsurv{0}=1$,\nwhich corresponds to the fact that nothing ever gets deleted from $D_{\\ell+1}(w)$,\nwhile $\\intfactsurv{1}=\\prob(\\mathrm{Bi}(r-1,1)\\geq 1)=1$,\ncorresponding to the fact that internal factor nodes have $r-1 \\ge 1$ children in the input graph\nand therefore also no nodes of $D_\\ell(w)$ will be deleted.\n\nWe assume both statements are true for~$t-1$ and aim to prove that they also hold for~$t$.\nWe first consider $u\\in D_{\\ell+1-2t}$ and let $X_u$ be the number of children of $u$ in $D^*_{\\ell+2-2t}$.\nObserve that the probability that $u$ survives in \\textnormal{\\texttt{CoreConstruct}}\\ is simply $\\prob(X_u\\geq 1)$.\n\nBy Proposition~\\ref{prop:distributionofnochildren},\nthe number of children of $u$ is asymptotically $\\mathrm{Po}(d)$, \nand by the induction hypothesis each child survives with probability $\\intfactsurv{t}+o(1)$ independently of each other.\nTherefore the asymptotic survival probability of $u$ is given by \n\\[\n\\prob\\left(\\mathrm{Bi}(\\mathrm{Po}(d),\\intfactsurv{t}+o(1))\\geq 1\\right)=\\prob\\left(\\mathrm{Po}(d(\\intfactsurv{t}+o(1)))\\geq 1\\right)=\\intvarsurv{t}+o(1),\n\\]\nby definition of $\\intvarsurv{t}$, as required for statement~(1).\nIndependence simply follows from the conditioning on $E_{w}(\\ell)$, which\nin particular means that $N_{\\le \\ell+1}(w)$ is a tree.\n\n\nSimilarly for $a\\in D_{\\ell-2t}(w)$ we define $X_a$ to be the number of children of $a$ which survive.\nSince $a$ is an internal factor node it has precisely $r-1$ children,\nand by statement~1 which we have just proved, each child survives with probability $\\intvarsurv{t}+o(1)$ independently.\nTherefore the probability that $a$ survives the \\textnormal{\\texttt{CoreConstruct}}\\ is\n$$\n\\Pr(\\mathrm{Bi}(r-1,\\intvarsurv{t}+o(1))\\ge 1) = \\intfactsurv{t+1}+o(1),\n$$\nas required for statement~(2). Again, independence simply follows from the conditioning on $E_{w}(\\ell)$.\n \\end{proof}\n\nA consequence of \nLemma~\\ref{lem:survivalprobs} is that, if $\\ell$ is large,\nthe distribution of the number of children of the root $w$ which survive $\\coredetector $ is almost identical\nto the claimed distributions in Theorem~\\ref{thm:factor:reducedcoredegs}.\nIn order to quantify this,\nfor two discrete random variables $X,Y$ taking values in $\\mathbb{N}$ we define the \\textit{total variation distance} as\n\\[\nd_{\\mathrm{TV}}(X,Y)\\coloneqq\\sum_{m\\in\\mathbb{N}}\\Big|\\prob(X=m)-\\prob(Y=m)\\Big|.\n\\]\n \n\n\\begin{corollary}\\label{cor:totalvariationdistance}\nThere exist $\\varepsilon=o(1)$ and $\\ell=\\ell(\\varepsilon)$ such that if we run $\\coredetector $ with input $\\ell$ and root $w\\in\\mathcal{V}\\cup\\mathcal{F}$, then:\n\\begin{enumerate}\n \\item If $w \\in \\mathcal{V}$, then\n \\[\n d_{\\mathrm{TV}}\\left(\\tilde d_1(w),\\widetilde \\mathrm{Po}(d\\hat \\rho_*)\\right) \\le d_{\\mathrm{TV}}\\left(d^*_1(w), \\mathrm{Po}(d\\hat \\rho_*)\\right)<\\varepsilon;\n \\]\n \\item If $w\\in\\mathcal{F}$, then\n \\[\n d_{\\mathrm{TV}}\\left(\\tilde d_1(w),\\widetilde \\mathrm{Bi}(r,\\rho_*)\\right)\\le d_{\\mathrm{TV}}\\left(d^*_1(w),\\mathrm{Bi}(r,\\rho_*)\\right)<\\varepsilon.\n \\]\n\\end{enumerate}\n\\end{corollary}\n\nLet us note that the second statement involves the $\\mathrm{Bi}(r,\\rho_*)$ distribution\nrather than $\\mathrm{Bi}(r-1,\\rho_*)$ since $w$ is the root, and therefore all $r$ of its\nneighbours are children rather than just $r-1$ children and one parent.\n\n\\begin{proof}\nWe will prove only the first statement; the proof of the second is very similar.\n\nNote that the first inequality follows directly from the definitions of $d^*_1(w)$ and the $\\widetilde\\mathrm{Po}$ distributions.\nThere are $2$ reasons why the second inequality is not quite immediate from Lemma~\\ref{lem:survivalprobs}:\n\\begin{enumerate}[label=\\textnormal{\\textbf{(R\\arabic*)}}]\n \\item\\label{item:eventereasonone}Lemma~\\ref{lem:survivalprobs} has the conditioning on the event $E_{w}(\\ell)$;\n \n \\item\\label{item:eventereasontwo} $\\intfactsurv{(\\ell+1)\/2}$ in Lemma~\\ref{lem:survivalprobs} has been replaced by $\\hat \\rho_*$.\n\n\\end{enumerate}\nTo overcome \\ref{item:eventereasonone},\nfor ease of notation we set $q_j\\coloneqq \\Pr\\left(d^*_1(w)=j\\right)$ and\\linebreak[4] $q_j'\\coloneqq\\Pr\\left(d^*_1(w)=j|E_{w}(\\ell)\\right)$ for each $j\\in \\mathbb{N}$,\nand define\n\\begin{align*}\nJ^+ & \\coloneqq\\left\\{j:q_j > q_j'\\right\\} \\qquad \\text{and}\\qquad \nJ^- \\coloneqq\\left\\{j:q_j < q_j'\\right\\}.\n\\end{align*}\nSince for any $w \\in \\mathcal{V} \\cup \\mathcal{F}$ we have\n$$\n\\sum_{j=0}^\\infty (q_j-q_j') = \\sum_{j=0}^\\infty q_j - \\sum_{j=0}^\\infty q_j' = 1-1=0,\n$$\nwe deduce that\n$$\n\\sum_{j=0}^\\infty \\left|q_j-q_j'\\right|\n = \\sum_{j \\in J^+} \\left(q_j-q_j'\\right) - \\sum_{j \\in J^-}\\left(q_j-q_j'\\right)\n= 2\\sum_{j \\in J^+}\\left(q_j-q_j'\\right).\n$$\nTherefore we have\n\\begin{align}\nd_{\\mathrm{TV}}\\left(d^*_1(w),d^*_1(w)|_{E_{w}(\\ell)}\\right)\n& = 2\\sum_{j\\in J^+} \\left(q_j-q_j'\\right) \\nonumber\\\\\n& \\leq 2\\cdot \\sum_{j \\in J^+} \\Pr\\left(d^*_1(w)=j\\right) - \\Pr\\left( (d^*_1(w)=j)\\capE_{w}(\\ell)\\right) \\nonumber\\\\\n& \\leq 2 \\cdot\\Pr\\left(\\overline{E_{w}(\\ell)}\\right) \\nonumber\\\\\n& \\leq 2 \\exp\\left(-\\Theta\\left(\\sqrt{\\log n}\\right)\\right) \\nonumber\\\\\n& \\le \\varepsilon\/3, \\label{eq:totalvariationconditioning}\n\\end{align}\nwhere we applied Lemma~\\ref{lem:eventE} in the penultimate line, and where the last line holds\nfor $\\varepsilon$ tending to $0$ sufficiently slowly.\n\nTo address \\ref{item:eventereasontwo} we note that if $\\ell = \\ell(\\varepsilon)$ is sufficiently large, then\n\\begin{equation}\\label{eq:totalvariationpoissons}\n d_{\\mathrm{TV}}\\left(\\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2}),\\mathrm{Po}(d\\hat \\rho_*)\\right)<\\varepsilon\/3\n\\end{equation}\nbecause $\\intfactsurv{t} \\xrightarrow{t\\to \\infty} \\hat \\rho_*$ by definition.\n\nTo complete the proof, observe that Lemma~\\ref{lem:survivalprobs}~\\ref{eq:probfactnodesurvives} implies that\n\\begin{align*}\nd_{\\mathrm{TV}}\\left(d^*_1(w)|_{E_{w}(\\ell)} \\, , \\, \\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2})\\right) =\nd_{\\mathrm{TV}}\\left(d^*_1(w)|_{E_{w}(\\ell)} \\, , \\, \\mathrm{Bi}\\left(\\mathrm{Po}(d),\\intfactsurv{(\\ell+1)\/2} \\right)\\right) \\le \\varepsilon\/3, \n\\end{align*}\nand combining this with~\\eqref{eq:totalvariationconditioning} and~\\eqref{eq:totalvariationpoissons},\nwe obtain\n\\begin{align*}\nd_{\\mathrm{TV}}\\left(d^*_1(w),\\mathrm{Po}(d\\hat \\rho_*)\\right)\n& \\le d_{\\mathrm{TV}}\\left(d^*_1(w),d^*_1(w)|_{E_{w}(\\ell)}\\right)\n\t+ d_{\\mathrm{TV}}\\left(d^*_1(w)|_{E_{w}(\\ell)},\\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2})\\right)\\\\\n\t& \\hspace{3cm}+ d_{\\mathrm{TV}}\\left(\\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2},\\mathrm{Po}(d\\hat \\rho_*)\\right)\\\\\n\t& \\le \\varepsilon\n\\end{align*}\nas required.\n\\end{proof}\nAs a further consequence of Corollary~\\ref{cor:totalvariationdistance}, we can asymptotically determine the expected degree distribution in $G_{\\ell}$.\n\\begin{corollary}\\label{cor:expectationvarfac}\nThere exist $\\varepsilon = o(1)$ and $\\ell=\\ell(\\varepsilon)$ such that for all $j \\in \\mathbb{N}$, \n\\[\\expec\\left(\\varpropl{j}{\\ell}\\right)=\\prob\\left(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j\\right)\\pm\\varepsilon,\\]\nand\n\\[\\expec\\left(\\factpropl{j}{\\ell}\\right)=\\prob\\left(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j\\right)\\pm\\varepsilon.\\]\n\\end{corollary}\n\n\n\n\\begin{proof}\nObserve that for any $j \\in \\mathbb{N}$ and any variable node $w$, by linearity of expectation and Corollary~\\ref{cor:Gldegalmosteq},\nfor some $\\varepsilon_1=o(1)$ we have\n$$\n\\expec\\left(\\varpropl{j}{\\ell}\\right)=\\prob(d_{G_{\\ell}}(w)=j) = \n\\prob\\left(\\tilde d_1(w)=j\\right) \\pm \\varepsilon_1\n$$\nFurthermore, Corollary~\\ref{cor:totalvariationdistance} implies that for some $\\varepsilon_2=o(1)$ we have\n$$\n\\prob\\left(\\tilde d_1(w)=j\\right) = \\prob\\left(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j\\right)\\pm\\varepsilon_2.\n$$\nCombining these two approximations and setting $\\varepsilon = \\varepsilon_1+\\varepsilon_2 = o(1)$ completes the proof of the first statement.\nThe second statement is proven similarly.\n\\end{proof}\nWe will show concentration of $\\varpropl{j}{\\ell}$ and $\\factpropl{j}{\\ell}$ around their expectations\nby an application of an Azuma-Hoeffding inequality (Lemma~\\ref{lem:Azuma}).\nWe will therefore define a vertex-exposure martingale\n(or, using the terminology of factor graphs, a variable-exposure martingale)\n $X_0,\\ldots,X_n$ on $G$, where\n$X_0=\\expec(\\varpropl{j}{\\ell})$ and $X_n=\\varpropl{j}{\\ell}$. This means that we reveal the variable nodes of $G$\nand their incident edges one by one,\nsuccessively revealing more and more information about the input graph.\nWe would like to show that this martingale satisfies a Lipschitz property so that we can apply Lemma~\\ref{lem:Azuma}.\nHowever, this Lipschitz property cannot be guaranteed in general.\nWe therefore restrict attention to a class of factor graphs on which the Lipschitz property does indeed hold.\n\n\\begin{definition}\\label{def:graphclass}\nLet $\\mathcal{G}$ be the class of factor graphs $G=G(H)$ of $r$-uniform hypergraphs $H$ on vertex set $[n]$\nsuch that no node has degree greater than $\\log n$ in $G$.\n\\end{definition}\nIt is important to observe that for fixed $r$ and for $p$ in the range we are considering\n(i.e. $\\Theta\\left(\\binom{n-1}{r-1}^{-1}\\right)$), the factor graph\n $G^r(n,p)$ is very likely to lie in $\\mathcal{G}$, so the restriction to this class is reasonable.\n\\begin{claim}\\label{claim:maxdeg}\nWe have \n\\[\\prob(G^r(n,p)\\in\\mathcal{G})=1-o(1).\\]\n\\end{claim}\nThe (standard) proof appears in Appendix~\\ref{app:maxdeg}.\n\nNow let $\\overline{G}=G^r(n,p)|_{\\mathcal{G}}$ be the factor graph of the $r$-uniform binomial random hypergraph $H^r(n,p)$\nconditioned on being in $\\mathcal{G}$. For each $i\\in\\mathbb{N}$ let $Z_i\\in\\{0,1\\}^{\\binom{[i-1]}{r-1}}$\nbe the sequence $(Y_{i,A})_{A \\in \\binom{[i-1]}{r-1}}$ of indicator random variables of the events\nthat there is a factor node of $\\overline{G}$ whose neighbours are $\\{i\\}\\cup A$\n(or equivalently that $\\{i\\}\\cup A$ forms an edge of $H^r(n,p)$).\nLet $f$ be a graph theoretic function.\nSince there is a one-to-one correspondence between the factor graph $\\overline{G}$ and its associated sequence $(Z_i)=(Z_i)_{i \\in [n]}$,\nwe can view the \nrandom variable $X\\coloneqq f(\\overline{G})$ as a function of the sequence $(Z_i)$ and we let $X_i\\coloneqq\\expec(X|Z_1,\\ldots,Z_i).$\nThis martingale can be seen as revealing variable nodes of $\\overline{G}$ one by one, and $X_i$ is\nthe expected value of $X$ conditioned on the information revealed after $i$ steps.\n\n\\begin{definition}\\label{lem:lipschitzproperty}\nA graph function $f$ is \\emph{$c$-variable-Lipschitz} if whenever two factor graphs $G$ and $G'$ differ at exactly one variable node\n(but any number of factor nodes),\nthen $|f(G)-f(G')|\\leq c$ holds.\n\\end{definition}\n\nThis is very similar to the standard notion of being $c$-vertex-Lipschitz -- indeed, it corresponds\nprecisely to the original hypergraph being $c$-vertex-Lipschitz -- and it is\na standard observation that being $c$-variable-Lipschitz\nimplies that the corresponding variable-exposure martingale satisfies $|X_i-X_{i-1}|\\leq c$,\nsee e.g.\\ Corollary 2.27 in~\\cite{JansonLuczakRucinskiBook}.\n\nFor the purposes of the Azuma-Hoeffding argument,\nwe will view $\\varpropl{j}{\\ell},\\factpropl{j}{\\ell}$ as graph functions---thus by the original\ndefinition we have e.g.\\ $\\varpropl{j}{\\ell} = \\varpropl{j}{\\ell}(G^r(n,p))$,\ni.e.\\ the graph function applied to the factor graph of the $r$-uniform binomial random hypergraph.\n\n\n \\begin{lemma}\nFor $\\ell\\in\\mathbb{N}$, the graph functions $\\varpropl{j}{\\ell},\\factpropl{j}{\\ell}$ are $c_{\\ell}$-variable-Lipschitz within the class $\\mathcal{G}$, where $c_{\\ell}=\\frac{2(\\log n)^{\\ell+1}}{n}$.\n \\end{lemma}\n \n \\begin{proof}\n We will show that $\\varpropl{j}{\\ell}$ is $c_\\ell$-variable-Lipschitz---the corresponding proof for $\\factpropl{j}{\\ell}$ is completely analogous.\n \n Suppose that $G,G' \\in \\mathcal{G}$ are two factor graphs which differ at exactly one variable node, say $v$,\n and let $A_j = A_j(G,\\ell)$ and $A_j'=A_j'(G',\\ell)$ be the sets of variable nodes of degree~$j$ in $G_\\ell$ and $G_{\\ell}'$, respectively.\n Since $G,G'$ differ only at $v$, any edges in the symmetric difference $G_\\ell \\Delta G_{\\ell}'$ must lie within\n distance $\\ell+1$ of $v$ in either $G$ or $G'$,\n and therefore any nodes in the symmetric difference $A_j \\Delta A_j'$ must lie within distance $\\ell+1$ of $v$ in either $G$ or $G'$.\nSince $G,G' \\in \\mathcal{G}$, and therefore have maximum degree at most $\\log n$,\n there are at most $(\\log n)^{\\ell+1}$ such nodes in each of $G,G'$, and\ntherefore $|A_j \\Delta A_j'| \\le 2(\\log n)^{\\ell+1}$. Since $\\varpropl{j}{\\ell}(G)=|A_j|\/n$ and $\\varpropl{j}{\\ell}(G')=|A_j'|\/n$,\nwe deduce that\n$$\n\\left|\\varpropl{j}{\\ell}(G)-\\varpropl{j}{\\ell}(G')\\right| \\le \\frac{2(\\log n)^{\\ell+1}}{n}\n$$\nas required.\n \\end{proof}\n \nNow for $j,\\ell \\in \\mathbb{N}$, let us define the events $B_j=B_j(\\ell) \\coloneqq \\left\\{\\left|\\varpropl{j}{\\ell} - \\expec\\left(\\varpropl{j}{\\ell}\\right)\\right| \\ge n^{-1\/3}\\right\\}$\nand $\\hat B_j=\\hat B_j(\\ell) \\coloneqq \\left\\{\\left|\\factpropl{j}{\\ell} - \\expec\\left(\\factpropl{j}{\\ell}\\right)\\right| \\ge n^{-1\/3}\\right\\}$.\n \n\\begin{lemma}\\label{lem:concentration}\nLet $G=G^r(n,p)$ be a random factor graph and $\\ell = o(\\log \\log n)$. Then\n\\[\n\\Pr\\left(\\bigcup_{j \\in \\mathbb{N}}\\left(B_j \\cup \\hat B_j\\right) \\right) = o(1).\n\\]\n\\end{lemma} \n\n\\begin{proof}\nFor convenience, we will prove that $\\Pr(\\bigcup_{j \\in \\mathbb{N}}B_j)=o(1)$---the proof of the corresponding\nstatement for the $\\hat B_j$ is almost identical.\n\nLet us define $B\\coloneqq \\bigcup_{j\\in \\mathbb{N}}B_j$, and observe that\n\\begin{align}\\label{eq:azumaconditioning}\n\\Pr(B) & = \\Pr(B \\cap \\{G \\in \\mathcal{G}\\}) + \\Pr(B \\cap \\{G \\notin \\mathcal{G}\\}) \\nonumber \\\\\n& \\le \\Pr(B | G \\in \\mathcal{G}) + \\Pr(G \\notin \\mathcal{G}) \\nonumber \\\\\n& \\le \\sum_{j \\in \\mathbb{N}} \\Pr(B_j | G \\in \\mathcal{G}) + o(1),\n\\end{align}\nwhere the last line follows by a union bound and Claim~\\ref{claim:maxdeg}.\n\nNow in order to bound the sum, let us first fix $j \\in \\mathbb{N}$.\nSince $\\ell=o(\\log\\log n)$, we have \n$(\\log n)^ {\\ell+1}=n^ {o(1)}$,\nand by Lemma~\\ref{lem:Azuma} with $c_{\\ell}=\\frac{2(\\log n)^ {\\ell+1}}{n}$ and $s=n^ {-1\/3}$,\nwe have\n\\begin{align*}\n \\prob\\left(\\left|\\varpropl{j}{\\ell}-\\expec\\left(\\varpropl{j}{\\ell}\\right)\\right|\\geq n^{-1\/3} \\; \\Bigg| \\; G \\in \\mathcal{G}\\right)\n &\\leq 2\\cdot \\exp\\left(-\\frac{n^{-2\/3}}{2\\cdot 2(\\log n)^{\\ell+1}\/n}\\right)< \\exp\\left(-n^{1\/2}\\right).\n\\end{align*}\nObserving that the degree of any node is (deterministically) bounded by $\\binom{n}{r-1}$,\nwe have\n$$\n\\sum_{j\\in \\mathbb{N}}\\Pr(B_j | G \\in \\mathcal{G}) \\le \\left(\\binom{n}{r-1}+1\\right)\\exp(-n^{1\/2}) = o(1).\n$$\nSubstituting this into~\\eqref{eq:azumaconditioning} completes the proof.\n\\end{proof}\nWe are now able to give the proof of Lemma~\\ref{lem:mainlemma1}.\n\\begin{proof}[Proof of Lemma~\\ref{lem:mainlemma1}]\nBy Corollary~\\ref{cor:expectationvarfac} (with $\\varepsilon\/2$ in place of $\\varepsilon$),\nif $\\ell = \\ell(\\varepsilon)$ is sufficiently large we have\n\\[\n\\expec\\left(\\varpropl{j}{\\ell}\\right)=\\prob\\left(\\widetilde\\mathrm{Po}\\left(d\\hat \\rho_*\\right)=j\\right)\\pm\\varepsilon\/2\n\\]\nfor any $j \\in \\mathbb{N}$.\nFurthermore, by\nLemma~\\ref{lem:concentration}, we have that whp\\ for all $j\\in\\mathbb{N}$\n\\[\n\\varpropl{j}{\\ell}=\\expec\\left(\\varpropl{j}{\\ell}\\right)+o(1) = \\expec\\left(\\varpropl{j}{\\ell}\\right) \\pm \\varepsilon\/2\n\\]\nfor $\\varepsilon$ tending to $0$ sufficiently slowly,\nand combining these two facts proves the lemma for variable nodes. The proof for factor nodes is essentially identical.\n\\end{proof}\n\n\n\n\n\n\n\\section{Subcriticality: Proof of Lemma~\\ref{lem:mainlemma2}}\\label{sec:prooflowerbound}\nOur goal in this section is to show that after some large number $\\ell$ rounds of the peeling process on $G^r(n,p)$ have been completed,\nwhp\\ very few nodes will be disabled in subsequent rounds (at most $\\varepsilon n$ for some $\\varepsilon=\\varepsilon(n) =o(1)$),\nthus proving Lemma~\\ref{lem:mainlemma2}.\n\n\\emph{\nLet us fix $\\ell,\\varepsilon_1$ as in Lemma~\\ref{lem:mainlemma1}, and for the rest of this section we will assume that\nthe high probability events of Lemma~\\ref{lem:mainlemma1} and Proposition~\\ref{prop:numberfactornodes}\nboth hold.\n}\n\nTo help intuitively describe the argument, let us suppose for simplicity that in round $\\ell$ exactly \\emph{one}\nnode $x_0$ is disabled and we consider the future effects of such a disabling.\nSince $x_0$ was disabled, it had degree at most one,\nand therefore there is at most one neighbour $x_1$ whose degree is decreased as a result.\nIf $x_1$ originally had degree two, it now has degree one and will therefore be disabled in round $\\ell+1$.\nContinuing in this way, we observe that we will never be disabling more than one node in any subsequent round.\nFurthermore, if we reach a node $x_i$ whose original degree was not exactly two, the peeling process stops\n(either without disabling this node, or once it has been disabled and no further nodes' degrees are decreased).\nHeuristically, it will not take long before we reach a node whose original degree was not exactly two---this\nis because Lemma~\\ref{lem:mainlemma1} implies in particular that a constant proportion\nof the nodes have degree at least three.\n\nOf course, in reality there may be more than one node disabled in round $\\ell$.\nThis slightly complicates matters because some node may receive the knock-on effects\nof more than one disabling in round $\\ell$, and therefore have its degree decreased by more than one.\nHowever, this will turn out to be no more than a technical nuisance.\n\nWe first need a result which states that almost any graph with a fixed (reasonable) degree sequence\nis approximately equally likely to be $G_\\ell$, the graph obtained from $G=G^r(n,p)$ after $\\ell$ rounds of the peeling process.\nTo introduce this result, we need some definitions.\n\n\\begin{definition}\nAn \\emph{$r$-duplicate} in a factor graph consists of two factor nodes and $r$ variable nodes which together\nform a copy of $K_{2,r}$.\n\\end{definition}\n\nObserve that an $r$-duplicate would correspond to a double-edge in an $r$-uniform hypergraph,\nwhich in our model cannot occur since the hypergraph must be simple.\nTherefore our factor graphs may not contain any $r$-duplicates.\nOn the other hand, a ``loop'', in the sense of an edge which contains the same vertex more than once,\nmust involve a double-edge in the corresponding factor graph. This motivates the following definition,\nwhich (roughly) describes when a factor graph corresponds to a simple hypergraph.\n\n\\begin{definition}\\label{def:rplain}\nWe say that a factor graph is \\emph{$r$-plain} if:\n\\begin{enumerate}\n\\item it contains no double-edge, i.e.\\ two edges between the same variable node and factor node;\n\\item it contains no $r$-duplicates.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\\begin{claim}\\label{claim:uniformity}\nSuppose $H_1,H_2$ are two $r$-plain factor graphs with common variable node set $\\mathcal{V}=\\mathcal{V}(H_1)=\\mathcal{V}(H_2)=[n]$\nand with factor node sets $\\mathcal{F}_1=\\mathcal{F}(H_1)$ and $\\mathcal{F}_2=\\mathcal{F}(H_2)$.\nSuppose further that there is a bijection $\\phi: \\mathcal{V} \\cup \\mathcal{F}(H_1) \\to \\mathcal{V} \\cup \\mathcal{F}(H_2)$\nsuch that\n\\begin{itemize}\n\\item $\\phi(\\mathcal{V})=\\mathcal{V}$;\n\\item $d_{H_2}(\\phi(v))=d_{H_1}(v)$ for all $v \\in \\mathcal{V} \\cup \\mathcal{F}(H_1)$.\n\\end{itemize}\nLet $G=G^r(n,p)$.\nThen \n\\[\n\\prob\\left(G_{\\ell}=H_1\\right)=\\prob\\left(G_{\\ell}=H_2\\right).\n\\]\n\\end{claim} \n\nThis claim is very similar to standard results for simple graphs or hypergraphs\n(see e.g.~\\cite{Molloy2005})\nand we defer the proof to Appendix~\\ref{sec:proofofuniformity}. However, we note that in our setting there \nis one subtle technical difficulty to overcome which does not appear in many other cases,\nnamely that given a factor graph $G$ such that $G_\\ell=H_1$,\nif we transform $G$ by changing $H_1$ to $H_2$ but otherwise leaving $G$ unchanged,\nwe need to show that\nthe resulting graph is indeed the factor graph of an $r$-uniform hypergraph, and in particular is $r$-plain.\n\n\nClaim~\\ref{claim:uniformity} tells us that the factor graph $G_\\ell$ after $\\ell$ rounds\nof the peeling process is uniformly random conditioned on its degree sequence and being $r$-plain.\nSince Lemma~\\ref{lem:mainlemma1} tells us the degree sequence quite precisely,\nthis is very helpful. We can change our point of view by saying that we first reveal\nthe degree sequence of $G_\\ell$ without revealing any of its edges,\nand subsequently we reveal edges only as required. More precisely, we consider the\n\\emph{configuration model}, in which each node is given half-edges based on its degree,\nand we generate a uniformly random perfect matching between the two classes of half-edges\n(at variable and factor nodes) conditioned on the\nresulting factor graph being $r$-plain.\nWe need to know that this conditioning is not too restrictive, i.e.\\ that the probability\nthat the resulting factor graph is $r$-plain is not too small. This will be stated\nin Proposition~\\ref{prop:simpleprob}, for which we first need some preliminaries.\nWe begin by observing that\n$G^r(n,p)$ does not have too many nodes of high degree.\n\n\\begin{definition}\\label{def:largedegsquare}\nGiven a function $\\omega=\\omega(n) \\to \\infty$, we say that a factor graph $H$\nhas property $\\widetilde{\\mathcal{D}}=\\widetilde{\\mathcal{D}}(\\omega,n)$\nif\n$$\n\\sum_{\\substack{v \\in \\mathcal{V}(H):\\ d(v)>\\omega}}\\ d(v)^2 = o(n).\n$$\nFurthermore given $\\varepsilon>0$ we say that $H$ has property\n$\\mathcal{D}=\\mathcal{D}(\\varepsilon,\\omega,n)$ if it satisfies property $\\widetilde{\\mathcal{D}}(\\omega,n)$\nand also satisfies the conclusion of Lemma~\\ref{lem:mainlemma1} (with this $\\varepsilon$).\n\\end{definition}\n\n\\begin{claim}\\label{claim:largedegsquare}\nFor any $\\omega\\xrightarrow{n \\to \\infty} \\infty$,\nwith high probability $G^r(n,p)$ has property $\\widetilde{\\mathcal{D}}(\\omega,n)$.\n\\end{claim}\n\\begin{proof}\nFor $k \\in \\mathbb{N}$, let us define $X_k$ to be the number of variable nodes of degree $k$\nand $X_{\\ge k}\\coloneqq \\sum_{j \\in \\mathbb{N}_{\\ge k}} X_j$.\nObserve that the expected degree of a vertex is $\\binom{n-1}{r-1}p = (1+o(1))d$.\nIt is a standard fact that the degrees of vertices are approximately\ndistributed as independent $\\mathrm{Po}(d)$ variables. More formally (though much weaker),\nit is an easy exercise in the second-moment method\nto prove that whp, for any $k\\in \\mathbb{N}$ we have $X_{\\ge k} \\le n\\cdot \\Pr(\\mathrm{Po}(2d)\\ge k)$ (we omit the details).\nWe therefore have\n\\begin{align*}\n\\sum_{v \\in \\mathcal{V}(H): d(v) \\ge \\omega} d(v)^2 = \\sum_{k \\ge \\omega} k^2 X_k\n& \\le n\\sum_{k \\ge \\omega} k^2 \\frac{e^{-2d}(2d)^k}{k!} = n \\cdot (1+o(1))\\omega^2\\frac{e^{-2d}(2d)^{\\omega}}{\\omega!} = o(n),\n\\end{align*}\nas required.\n\\end{proof}\n\nNote that if $\\widetilde{\\mathcal{D}}$ holds in a factor graph $G$, then it also holds\nin any subgraph of $G$, and in particular in $G_{\\ell}$, the factor graph obtained\nafter $\\ell$ steps of the peeling process. Together with Lemma~\\ref{lem:mainlemma1},\nthis shows that, with $\\ell$ and $\\varepsilon$ as in given in that lemma and any $\\omega \\to \\infty$,\nsetting $G = G^r(n,p)$,\nwith high probability $G_\\ell$ satisfies $\\mathcal{D}(\\varepsilon,\\omega,n)$.\n\nLet us observe further that $\\mathcal{D}$ is a property that depends only on the\ndegree sequences of variable and factor nodes of the graph, and therefore with a slight\nabuse of terminology we may also say that it holds in a factor graph with half-edges,\nwhere we have not yet determined which half-edges will be matched together.\n\n\n\n\n\\begin{proposition}\\label{prop:simpleprob}\nLet $\\varepsilon = o(1)$ and\nsuppose that $n$ variable nodes and $m=(1+o(1))\\frac{dn}{r}$ factor nodes are given half-edges\nin such a way that property $\\mathcal{D}(\\varepsilon,\\varepsilon^{-1\/4},n)$ holds.\nSuppose also that the total numbers of half-edges at factor nodes and at variable nodes are equal,\nand that we construct a uniformly random perfect matching between these two sets of half-edges.\nThen there exists a constant $c_0>0$ (independent of $\\varepsilon,n$)\nsuch that for sufficiently large $n$ the probability that the resulting factor graph is\n$r$-plain is at least $c_0$.\n\\end{proposition}\n\n\n\nThe proof of Proposition~\\ref{prop:simpleprob} is a standard exercise\nin applying the method of moments to determine the asymptotic distribution\nof the number of double edges and $r$-duplicates -- we omit the details.\nThe proposition states in particular that we may condition on the\nresulting graph being $r$-plain without skewing the distribution of the matching too much.\nMore precisely, any statements that are true with high probability for the uniformly random perfect matching\nare also true with high probability under the condition that the resulting graph is simple.\nTherefore in what follows, for simplicity we will suppress this conditioning.\n\n\\begin{definition}[Change process]\\label{def:changeprocess}\nWe will track the changes that the peeling process makes after reaching round $\\ell$\nby revealing information a little at a time as follows.\n\\begin{itemize}\n\\item Reveal the degrees of all nodes.\n\\item While there are still nodes of degree one, pick one such node $x_0$.\n\\begin{itemize}\n\\item Reveal its neighbour $x_1$, delete the edge $x_0x_1$ and update the degrees of $x_0,x_1$.\n\\item If $x_1$ now has degree one, continue from $x_1$; otherwise find a new $x_0$ (if there is one).\n\\end{itemize}\n\\end{itemize}\n\\end{definition}\n\n\n\\noindent In other words, we track the changes in a depth-first search manner (rather than the breadth-first\nview of considering rounds of the peeling process). We call this the \\emph{change process}.\n\nObserve that we only reveal edges one at a time (just before deleting them).\nThe following claim is simple given Lemma~\\ref{lem:mainlemma1}, but is the essential heart of our proof.\nRecall that $\\varepsilon_2 \\coloneqq \\sqrt{\\varepsilon_1}$ as defined in Lemma~\\ref{lem:mainlemma2}.\n\n\n\\begin{claim}\\label{claim:terminationprob}\nLet $G'$ be any graph obtained from $G_\\ell$ by deleting at most $\\varepsilon_2 n$ edges.\nThen when revealing the second endpoint of any half-edge, the probability of revealing a node of degree at least three\nis at least\n$$\n\\min\\left\\{\\frac{(d\\hat \\rho_*)^2 \\exp(-d\\hat \\rho_*)}{2} , \\frac{(r-1)(r-2)\\rho_*^2(1-\\rho_*)^{r-3}}{2}\\right\\} - 20\\varepsilon_2.\n$$\nIn particular there exists a constant $c=c(r,d)>0$ such that this probability is at least $c$.\n\\end{claim}\nWe defer the proof to Appendix~\\ref{sec:proofofterminationprob}.\n\n\\noindent This claim tells us that, provided we have not deleted too many edges so far, there is a reasonable probability\nof revealing a node of degree at least three, which blocks the continued propagation of any deletions.\n\nLet $c=c(r,d)$ be as in Claim~\\ref{claim:terminationprob}\nand let us set $\\delta_1\\coloneqq\\varepsilon_1^{3\/4}$.\nWe now define an abstract branching process\nwhich will provide an upper coupling on the change process starting from $G_\\ell$.\n\n\\begin{definition}\nLet $\\mathcal{T}$ be a branching process which begins with $\\delta_1n$ vertices in generation~$0$,\nand in which each vertex independently has a child with probability $1-c$, and otherwise has no children.\n\\end{definition}\n\n\n\n\\begin{proposition}\\label{prop:changecoupling}\nThe process $\\mathcal{T}$ can be coupled with the change process in such a way that,\nif both processes are run until one of the stopping conditions\n\\begin{itemize}\n\\item $\\mathcal{T}$ has reached size at least $\\varepsilon_2 n$;\n\\item $\\mathcal{T}$ has died out,\n\\end{itemize}\nis satisfied, then $\\mathcal{T}$ forms an upper coupling on the change process.\n\\end{proposition}\n\n\\begin{proof}\nWe first need to show that whp\\ we make at most $\\delta_1 n$ changes in round $\\ell+1$ of the peeling process.\nSince we have assumed that the high probability statement of Lemma~\\ref{lem:mainlemma1} holds, we have\n$\\varpropl{j}{\\ell}=\\prob(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j)\\pm\\varepsilon_1$ and $\\factpropl{j}{\\ell}=\\prob(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j)\\pm\\varepsilon_1$.\nBy the definition of the peeling process (Definition~\\ref{def:peeling}) the only change we make when moving from $G_{\\ell}$ to $G_{\\ell+1}$ is that we disable all nodes of degree one.\nThe proportion of such variable and factor nodes in $G_{\\ell}$ is $\\varpropl{1}{\\ell}$ and $\\factpropl{1}{\\ell}$ respectively.\nRecalling that $\\prob(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=1) = \\prob(\\widetilde\\mathrm{Bi}(r,\\rho_*)=1) =0$,\nthis immediately implies that at most\n$\\varepsilon_1(m+n) \\le \\delta_1 n$ nodes are disabled in round $\\ell+1$ of the peeling process,\nand these disablings represent the first nodes of the change process.\n\n\nNow the proposition follows directly from the observation that in the change process,\na node only has at most one incident edge deleted (if it has degree one), and\ntherefore at most one neighbour is revealed, along with Claim~\\ref{claim:terminationprob},\nwhich implies that the probability of not causing any further changes is at least $c$.\nThe first stopping condition ensures that we have deleted at most $\\varepsilon_2 n$ edges,\nand therefore the assumptions of Claim~\\ref{claim:terminationprob} are indeed satisfied.\n\\end{proof}\n\nIn view of Proposition~\\ref{prop:changecoupling}, it is enough to prove that whp\\ the\nbranching process $\\mathcal{T}$ dies out (i.e.\\ fulfills the second stopping condition)\nbefore reaching size $\\varepsilon_2 n$.\n\n\\begin{proposition}\\label{prop:changebranchsmall}\nWhp\\ $\\mathcal{T}$ contains at most $\\varepsilon_2 n$ vertices.\n\\end{proposition}\n\n\\begin{proof}\nIn order to reach size $\\varepsilon_2 n$, the first (at most) $\\varepsilon_2 n$ vertices of the process\nwould have to have a total of at least $(\\varepsilon_2-\\delta_1)n$ children,\nwhich, by Lemma~\\ref{lem:chernoffbounds}, occurs with probability at most\n\\begin{align*}\n\\Pr\\left(\\mathrm{Bi}(\\varepsilon_2 n,1-c)\\ge (\\varepsilon_2-\\delta_1)n\\right) & \\le 2\\cdot \\exp\\left(-\\frac{(c\\varepsilon_2-\\delta_1)^2 n^2}{2\\varepsilon_2 n + 2(c\\varepsilon_2-\\delta_1)n\/3}\\right)\\\\\n& \\le 2\\cdot \\exp\\left(-\\frac{(c\\varepsilon_2\/2)^2n}{3\\varepsilon_2}\\right) =o(1)\n\\end{align*}\n(since $\\varepsilon_2 n = \\sqrt{\\varepsilon_1}n \\to \\infty$)\nas required.\n\\end{proof}\n\nWe can now complete the proof of Lemma~\\ref{lem:mainlemma2}.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:mainlemma2}]\nProposition~\\ref{prop:changecoupling} implies that the probability that at least $\\varepsilon_2 n$\nfurther nodes are disabled after round $\\ell$ in the peeling process is at most the probability\nthat $\\mathcal{T}$ reaches size $\\varepsilon_2 n$. But this event has probability $o(1)$ by\nProposition~\\ref{prop:changebranchsmall}.\n\nThis proves that whp\\ at most $\\varepsilon_2 n$ further nodes are disabled\nafter round $\\ell$ in the peeling process. To show the remaining two statements,\nobserve that\nsince disabling a node deletes at most one edge, and therefore changes the degree of at most one variable node\nand at most one factor node, at most $\\varepsilon_2 n$ nodes of each type will have their degree changed.\nIt follows immediately that for any $j \\in \\mathbb{N}$ we have $\\varprop{j} = \\varpropl{j}{\\ell} \\pm \\varepsilon_2$,\nwhile similarly\n$$\n\\factprop{j} = \\factpropl{j}{\\ell} \\pm \\frac{\\varepsilon_2 n}{m} = \\factpropl{j}{\\ell} \\pm \\frac{2\\varepsilon_2 r}{d},\n$$\nwhere we have used the fact that whp\\ $\\frac{n}{m} \\le \\frac{2r}{d}$ by Proposition~\\ref{prop:numberfactornodes}.\n\\end{proof}\n\n\n\n\n\\section{Concluding remarks}\\label{sec:concluding}\n\n\\subsection{Upper bound on $L_P,L_C$}\nTheorem~\\ref{thm:mainresultcycle} and Corollary~\\ref{cor:upperboundlongestpath} state that whp\\ $L_P$ and $L_C$,\nthe length of the longest loose path and longest loose cycle respectively in $H^r(n,p)$,\nsatisfy $L_P,L_C\\leq(\\min\\{\\beta,\\gamma\\}+o(1))\\cdot n$, but which of $\\beta,\\gamma$ is smaller?\nRecall that both $\\beta$ and $\\gamma$ are functions of $d$ and $r$, with $\\beta=\\gamma =0$ for $d 0$ such that\n \\begin{align}\n \\label{eq:globalLF\n |F(u)-F(v)|\n \\leq&~\n L_{F}|u-v|,\n \\quad u,v \\in H,\n \\\\\n \\label{eq:auxiliaryassumptionFuH1\n |F(u)|_{\\dot{H}^{1}}\n \\leq&~\n L(1+|u|_{\\dot{H}^{1}}),\n \\quad u \\in \\dot{H}^{1},\n \\\\\n \\label{eq:auxiliaryassumption\n |F'(u)v|_{\\dot{H}^{2}}\n \\leq&~\n L\\big(1+|u|_{\\dot{H}^{2}}^{2}\\big)|v|_{\\dot{H}^{2}},\n \\quad u,v \\in \\dot{H}^{2}.\n \\end{align}\n\\end{Assumption}\n\n Concerning Assumptions \\ref{ass:AQ} and \\ref{ass:F}, for example, one may take $q_{i} = \\lambda_{i}^{-\\delta},i \\in \\mathbb{N}^{+}$\n for some $\\delta \\in (\\frac{3}{2},2]$\n and $F \\colon H \\to H$ a Nemytskij operator defined by\n $\n F(u)(\\xi) := f(u(\\xi)),\\xi \\in (0,1), u \\in H $\n with $f \\in C_{b}^{3}(\\mathbb{R})$.\n\n\n\n\n\nFrom \\cite[Theorem 7.2]{da2014stochastic}, we know that \\eqref{eq:SPDE} admits a unique mild solution.\n\\begin{Proposition}\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then \\eqref{eq:SPDE} admits a unique mild solution $\\{X_{x}^{\\varepsilon}(t)\\}_{t \\geq 0}$ given by\n \n \\begin{equation}\\label{eq:SPDEmildsolution\n X_{x}^{\\varepsilon}(t)\n =\n E(t)x\n +\n \\int_{0}^{t} E(t-s) F(X_{x}^{\\varepsilon}(s)) \\diff{s}\n +\n \\varepsilon \\int_{0}^{t} E(t-s)Q^{\\frac12} \\diff{W(s)},\n \\quad t \\geq 0,\\quad\\P\\text{-a.s.}\n \\end{equation}\n Moreover, it belongs to $L^{p}(\\Omega;C([0,T];H))$ for any $p \\geq 1$ and $T > 0$.\n\\end{Proposition}\n\n\n\n\n\n\n\n\n\n\n\nTo formulate the LDPs for the solution of \\eqref{eq:SPDE}, we introduce some preliminaries upon the theory of large deviations; see \\cite[Chapter 1]{dembo2009large}. In what follows, let $(\\mathcal{U},\\rho^{\\mathcal{U}})$ be a Polish space and $\\mathcal{B}(\\mathcal{U})$ its Borel $\\sigma$-field.\n\n\n\n\n\n\\begin{Definition\n A rate function $I$ is a lower semi-continuous mapping $I \\colon \\mathcal{U} \\to [0,+\\infty]$ (such that for all $\\alpha \\geq 0$, the level set $K_{I}(\\alpha) := \\{u \\in \\mathcal{U} : I(u) \\leq \\alpha\\}$ is a closed subset of $\\mathcal{U}$). A good rate function is a rate function for which all the level sets $K_{I}(\\alpha)$ are compact subsets of $\\mathcal{U}$.\n\\end{Definition}\n\n\n\n\n\n\n\\begin{Definition\n\\label{def:generaldefLDP\n Let $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ be a family of probability measures on $(\\mathcal{U},\\mathcal{B}(\\mathcal{U}))$. We say that $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfies an LDP on $\\mathcal{U}$ with the rate $\\frac{1}{\\varepsilon}$ and the rate function $I$ if\n \\begin{equation}\\label{eq:LDPclassical\n -\\inf_{u \\in U^{o}} I(u)\n \\leq\n \\liminf_{\\varepsilon \\to 0} \\varepsilon \\log \\mu^{\\varepsilon}(U)\n \\leq\n \\limsup_{\\varepsilon \\to 0} \\varepsilon \\log \\mu^{\\varepsilon}(U)\n \\leq\n -\\inf_{u \\in \\overline{U}} I(u),\n \\quad U \\in \\mathcal{B}(\\mathcal{U}),\n \\end{equation}\n where $U^{o}$ denotes the interior of $U$,\n and $\\overline{U}$ the closure of $U$.\n\\end{Definition}\n\n\n\n\n\n\n\n\nThe following proposition about Freidlin--Wentzell exponential estimates gives an equivalent characterization of LDP; see \\cite[Chapter 3]{freidlin1984random}. \n\n\n\n\n\n\\begin{Proposition}\n If $(\\mathcal{U},\\rho^{\\mathcal{U}})$ is a Polish space and $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ is a family of probability measures on $(\\mathcal{U},\\mathcal{B}(\\mathcal{U}))$, then $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfying an LDP on $\\mathcal{U}$ with the rate $\\frac{1}{\\varepsilon}$ and the good rate function $I$ is equivalent to the following statements:\n \\begin{enumerate}\n \\item [(i)] compact level set: the level set $K_{I}(\\alpha) := \\{u \\in \\mathcal{U} : I(u) \\leq \\alpha\\}$ is compact for every $\\alpha \\geq 0$;\n\n \\item [(ii)] lower bound: for any $u \\in \\mathcal{U}$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n $$\\mu^{\\varepsilon}(\\{v \\in \\mathcal{U}:\\rho^{\\mathcal{U}}(u,v) < \\delta\\})\n \\geq \\exp\\Big(-\\frac{I(u)+\\gamma}{\\varepsilon}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0};$$\n\n \\item [(iii)] upper bound: for any $\\alpha > 0$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n $$\\mu^{\\varepsilon}(\\{v \\in \\mathcal{U}:\n \\rho^{\\mathcal{U}}(v,K_{I}(\\alpha)) \\geq \\delta\\})\n \\leq \\exp\\Big(-\\frac{\\alpha-\\gamma}{\\varepsilon}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0}.$$\n \\end{enumerate}\n\\end{Proposition}\n\n\n\n\n\n\nRegarding $X_{x}^{\\varepsilon}$ as a $C([0,T];H)$-value random variable for each $\\varepsilon > 0$, the uniform LDP of sample paths of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ can be summarized as follows; see \\cite{peszat1994large} for more details.\n\\begin{Theorem}\n\\label{th:exactsoluLDP\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H)$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $I_{0,T}^{x}$ defined by\n \\begin{equation}\\label{eq:I0TxzI0Txz\n I_{0,T}^{x}(z)\n =\n \\frac{1}{2}\n \\inf\\{|\\varphi|_{L^2(0,T;H)}^{2}:\n \\varphi \\in L^2(0,T;H),\n z_{0,x}^{\\varphi} = z\\},\n \\quad z \\in C([0,T];H),\n \\end{equation}\n where $z_{0,x}^{\\varphi}$ is the mild solution of the skeleton equation\n \\begin{equation}\\label{eq:skeletoneqofSPDE\n \\frac{\\dif{z_{0,x}^{\\varphi}(t)}}{\\dif{t}}\n =\n Az_{0,x}^{\\varphi}(t)\n + F(z_{0,x}^{\\varphi}(t))\n + Q^{\\frac12}\\varphi(t),\n \\quad t \\in (0,T],\n \\quad z_{0,x}^{\\varphi}(0) = x.\n \\end{equation}\n\\end{Theorem}\n\n\n\n\nTo ensure the existence of invariant measures for \\eqref{eq:SPDE}, we need the following assumption on dissipativity.\n\n\n\\begin{Assumption\n\\label{ass:LFleqlambda1\n Assume $F(0) = 0$ and $L_{F} < \\lambda_{1}$, where $L_{F}$ is the Lipschitz constant of $F$ given by \\eqref{eq:globalLF} and $\\lambda_{1}$ is the smallest eigenvalue of $-A$.\n\\end{Assumption}\n\n\n\n\n\n\n\n\n\n\nThis assumption together with\n$-Ae_{i} = \\lambda_{i}e_{i}, i \\in \\mathbb{N}^{+}$\nshows\n\\begin{equation}\\label{eq:dissipativecond\n \\langle Au + F(u), u \\rangle\n \\leq -c|u|^{2},\n \\quad u \\in \\dot{H}^{2}\n\\end{equation}\nwith $c:= \\lambda_{1}-L_{F} > 0$.\nIt follows from \\cite{cerrai2005largeinvariant} that there exists $\\{t_{i}\\}_{i \\in \\mathbb{N}^{+}} \\uparrow +\\infty$ (possibly depending on $\\varepsilon$) such that the sequence of probability measures $\\{\\mu_{t_{i}}^{\\varepsilon}\\}_{i \\in \\mathbb{N}^{+}}$, defined by\n $$\n \\mu_{t_{i}}^{\\varepsilon}(B)\n :=\n \\frac{1}{t_{i}} \\int_{0}^{t_{i}}\n \\P\\big(X_{0}^{\\varepsilon}(s) \\in B\\big)\n \\diff{s},\n \\quad B \\in \\mathcal{B}(H),\n $$\nconverges weakly to some probability measure $\\mu^{\\varepsilon}$ on $(H,\\mathcal{B}(H))$, which is invariant for \\eqref{eq:SPDE}. Moreover, the following theorem shows that $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ obeys an LDP on $H$; see, e.g., \\cite{cerrai2005largeinvariant}. \n\n\n\n\n\\begin{Theorem}\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfies an LDP on $H$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $V$ defined by\n \\begin{equation}\\label{eq:invaLdpratefun\n V(u)\n =\n \\inf\\{I_{0,T}^{0}(z):\n T>0, z \\in C([0,T];H), z(0) = 0, z(T) = u\\},\n \\quad u \\in H.\n \\end{equation}\n\\end{Theorem}\n\n\n\n\n\n\n\n\\section{Spatial discretization and its LDPs}\n\\label{sec:spatial\n\n\nHere we discretize \\eqref{eq:SPDE} in space by the spectral Galerkin method and present its uniform LDP of sample paths in Subsection \\ref{sec:SGM}.\nSubsection \\ref{subsec:semiweakasympre} obtains the weakly asymptotical preservation for the LDP of sample paths of the original equation. After establishing the LDP of invariant measures of the spatial discretization in Subsection \\ref{subsec:semiinvameaLDPs}, we prove its weakly asymptotical preservation for the LDP of invariant measures of the original equation in Subsection \\ref{sebsec:semiinvameaLDPasy}.\n\n\n\\subsection{Spectral Galerkin method}\\label{sec:SGM\n\nFor each $n \\in \\mathbb{N}^{+}$, we define $H_{n} := \\text{span}\\{e_{1},\\ldots,e_{n}\\} \\subset H$, the projection operator $P_{n} \\colon H \\to H_{n}$ by $P_{n}u = \\sum_{i=1}^{n}\\langle e_{i},u \\rangle e_{i}$ for every $u \\in H$, and\n$$\n y := P_{n}x,\n \\quad F_{n} := P_{n}F,\n \\quad Q_{n}^{\\frac12} := P_{n}Q^{\\frac12},\n \\quad W_{n} := P_{n}W.\n$$\nThe spectral Galerkin method applied to \\eqref{eq:SPDE} is given by\n\\begin{equation}\\label{eq:SPDEsemi\n \\diff{X_{y}^{\\varepsilon,n}(t)}\n =\n \\big(A_{n}X_{y}^{\\varepsilon,n}(t)\n +\n F_{n}(X_{y}^{\\varepsilon,n}(t))\\big)\\diff{t}\n +\n \\varepsilon Q_{n}^{\\frac12}\\diff{W_{n}(t)},\n \\quad t > 0,\n \\quad X_{y}^{\\varepsilon,n}(0) = y,\n\\end{equation}\nwhere $A_{n} \\colon H_{n} \\to H_{n}$ is defined by $A_{n} := P_{n}A$ and generates a $C_{0}$-semigroup $\\{E_{n}(t) := e^{tA_{n}}\\}_{t \\geq 0}$ on $H_{n}$.\nUnder Assumptions \\ref{ass:AQ} and \\ref{ass:F}, Theorem 4.5.3 in \\cite{kloeden1992numerical} ensures that \\eqref{eq:SPDEsemi} admits a unique mild solution\n\\begin{equation}\\label{eq:SPDEsemisolu\n X_{y}^{\\varepsilon,n}(t)\n =\n E_{n}(t)y\n +\n \\int_{0}^{t} E_{n}(t-s) F_{n}(X_{y}^{\\varepsilon,n}(s)) \\diff{s}\n +\n \\varepsilon\n \\int_{0}^{t} E_{n}(t-s)Q_{n}^{\\frac12} \\diff{W_{n}(s)},\n \\quad t \\geq 0.\n\\end{equation}\nMoreover, the solution process $\\{X_{y}^{\\varepsilon, n}(t)\\}_{t \\in [0,T]}$ belongs to $L^{p}(\\Omega;C([0,T];H_{n}))$ for any $p \\geq 1$ and $T > 0$. To show $\\{X_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ satisfying an LDP on $C([0,T];H_{n})$, we first note that the skeleton equation corresponding to \\eqref{eq:SPDEsemi} is given by\n\\begin{equation}\\label{eq:SPDEsemiskeletoneq\n \\frac{\\dif{z_{0,y}^{n,\\psi}(t)}}{\\dif{t}}\n =\n A_{n}z_{0,y}^{n,\\psi}(t) + F_{n}(z_{0,y}^{n,\\psi}(t)) + Q_{n}^{\\frac12}\\psi(t),\n \\quad t \\in (0,T],\n \\quad z_{0,y}^{n,\\psi}(0) = y\n\\end{equation}\nwith $\\psi \\in L^{2}(0,T;H_{n})$. According to\n\\cite[Theorem 1.1]{peszat1994large}, we know that \\eqref{eq:SPDEsemiskeletoneq} admits a unique mild solution $z_{0,y}^{n,\\psi} \\in C([0,T];H_{n})$, given by\n\\begin{equation}\\label{eq:skeletoneqspatial\n z_{0,y}^{n,\\psi}(t)\n =\n E_{n}(t)y\n +\n \\int_{0}^{t} E_{n}(t-s)F_{n}(z_{0,y}^{n,\\psi}(s)) \\diff{s}\n +\n \\int_{0}^{t} E_{n}(t-s)Q_{n}^{\\frac12}\\psi(s) \\diff{s},\n \\quad t \\in [0,T].\n\\end{equation}\nWith this, we define $I_{0,T}^{n,y} \\colon C([0,T];H_{n}) \\to [0,+\\infty]$ by\n\\begin{equation}\\label{eq:SPDEsemiratefun\n I_{0,T}^{n,y}(z)\n =\n \\frac{1}{2}\\inf\\{|\\psi|_{L^2(0,T;H)}^{2}:\n \\psi \\in L^2(0,T;H_{n}), z_{0,y}^{n,\\psi} = z\\},\n \\quad z \\in C([0,T];H_{n}).\n\\end{equation}\n\n\nFollowing the approach in \\cite{peszat1994large}, one can prove the uniform LDP of sample paths of $\\{X_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$.\n\n\n\n\\begin{Theorem}\n\\label{thm:semisolutionLDP\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then $\\{X_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H_{n})$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $I_{0,T}^{n,y}$ given by \\eqref{eq:SPDEsemiratefun}, i.e.,\n\n\n \\begin{enumerate}\n \\item [(i)] compact level set: for any $T > 0$, $n \\in \\mathbb{N}^{+}$ and $y \\in H_{n}$, the level set $K_{0,T}^{n,y}(\\alpha) := \\{z \\in C([0,T];H_{n}): I_{0,T}^{n,y}(z) \\leq \\alpha\\}$ is compact for every $\\alpha \\geq 0$;\n \\item [(ii)] uniform lower bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_0 > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$ and $z \\in K_{0,T}^{n,y}(\\alpha)$,\n \\begin{equation}\\label{eq:semisoluldplower\n \\P(|X_{y}^{\\varepsilon,n}-z|_{C([0,T];H)} < \\delta)\n \\geq\n \\exp\\Big(-\\frac{I_{0,T}^{n,y}(z)+\\gamma}{\\varepsilon^2}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0};\n \\end{equation}\n \\item [(iii)] uniform upper bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_0 > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$,\n \\begin{equation}\\label{eq:semisoluldpupper\n\t \\P(\\rho^{C([0,T];H)}\n (X_{y}^{\\varepsilon,n},K_{0,T}^{n,y}(\\alpha))\n \\geq \\delta)\n\t \\leq\n\t \\exp\\Big(-\\frac{\\alpha-\\gamma}{\\varepsilon^2}\\Big),\n\t \\quad \\varepsilon \\leq \\varepsilon_{0},\n \\end{equation}\n where $\\rho^{C([0,T];H)}(z,U) := \\inf\\limits_{z' \\in U}|z-z'|_{C([0,T];H)}, z \\in C([0,T];H), U \\subset C([0,T];H)$.\n \\end{enumerate}\n\\end{Theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Weakly asymptotical preservation for LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$}\n\\label{subsec:semiweakasympre\n\n\n\nThis part aims to estimate the error between the rate functions $I_{0,T}^{n,y}$ and $I_{0,T}^{x}$. To this end, we first simplify the expression of $I_{0,T}^{x}$, which relies on the spatial regularity of the solution of the skeleton equation \\eqref{eq:skeletoneqofSPDE}.\n\n\n\\begin{Lemma}\n\\label{lem:spatialregularity\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold and let $\\{z_{0,x}^{\\varphi}(t)\\}_{t \\in [0,T]}$ with $\\varphi \\in L^{2}(0,T;H)$ be given by \\eqref{eq:skeletoneqofSPDE}. If\n $x \\in \\dot{H}^{2}$, then $z_{0,x}^{\\varphi}(t) \\in \\dot{H}^{2},t \\in [0,T]$ and there exists $C > 0$ such that\n \\begin{equation}\\label{eq:38\n |z_{0,x}^{\\varphi}(t)|_{\\dot{H}^{2}}\n \\leq\n C(1+|x|_{\\dot{H}^{2}}),\n \\quad t \\in [0,T].\n \\end{equation}\n\\end{Lemma}\n\n\\begin{proof}\n Using $|E(t)|_{\\mathcal{L}(H)} \\leq e^{-\\lambda_{1}t} \\leq 1,t \\geq 0$, \\eqref{eq:globalLF} and the H\\\"{o}lder inequality\n leads to\n \\begin{equation*}\n \\begin{split}\n &~|z_{0,x}^{\\varphi}(t)|\n \\leq\n |E(t)x|\n +\n \\int_{0}^{t} |E(t-s)F(z_{0,x}^{\\varphi}(s))| \\diff{s}\n +\n \\int_{0}^{t} |E(t-s)Q^{\\frac12}\\varphi(s)| \\diff{s}\n \\\\\\leq&~\n |x|\n +\n C\\int_{0}^{t} (1+|z_{0,x}^{\\varphi}(s)|) \\diff{s}\n +\n |Q^{\\frac12}|_{\\mathcal{L}(H)} \\Big(\\int_{0}^{t} 1^{2} \\diff{s}\\Big)^{\\frac{1}{2}}\n \\Big(\\int_{0}^{t} |\\varphi(s)|^{2} \\diff{s}\\Big)^{\\frac{1}{2}}\n \\\\\\leq&~\n |x| + CT\n +\n \\sqrt{T}|Q^{\\frac12}|_{\\mathcal{L}(H)}\n |\\varphi|_{L^{2}(0,T;H)}\n +\n C\\int_{0}^{t} |z_{0,x}^{\\varphi}(s)| \\diff{s}.\n \\end{split}\n \\end{equation*}\n The Gronwall inequality implies\n \\begin{equation}\\label{eq:39393939\n |z_{0,x}^{\\varphi}(t)|\n \\leq\n C(1+|x|), \\quad t \\in [0,T].\n \\end{equation}\n Similarly, we use $|(-A)^{\\gamma}E(t)|_{\\mathcal{L}(H)}\n\t \\leq Ct^{-\\gamma}, t > 0,\\gamma \\geq 0$ to get\n \\begin{align*}\n &|z_{0,x}^{\\varphi}(t)|_{\\dot{H}^{1}}\n \\leq\n |E(t)x|_{\\dot{H}^{1}}\n +\n \\int_{0}^{t} |E(t-s)F(z_{0,x}^{\\varphi}(s))|_{\\dot{H}^{1}} \\diff{s}\n +\n \\int_{0}^{t} |E(t-s)Q^{\\frac12}\\varphi(s)|_{\\dot{H}^{1}} \\diff{s}\n \\\\\\leq&\n |x|_{\\dot{H}^{1}}\n +\n C\\int_{0}^{t} |(-A)^{\\frac{1}{2}}E(t-s)|_{\\mathcal{L}(H)}\n |F(z_{0,x}^{\\varphi}(s))| \\diff{s}\n +\n |(-A)^{\\frac{1}{2}}Q^{\\frac12}|_{\\mathcal{L}(H)}\n \\int_{0}^{t} |\\varphi(s)| \\diff{s}\n \\\\\\leq&\n |x|_{\\dot{H}^{1}} + C(1+|x|)\n +\n \\sqrt{T}\n |(-A)^{\\frac{1}{2}}Q^{\\frac12}|_{\\mathcal{L}_{2}(H)}\n |\\varphi|_{L^{2}(0,T;H)}\n \\\\\\leq&\n C(1+|x|_{\\dot{H}^{1}}).\n \\end{align*}\n In the same way,\n \\begin{align*}\n |z_{0,x}^{\\varphi}(t)|_{\\dot{H}^{2}}\n \\leq&~\n |E(t)x|_{\\dot{H}^{2}}\n +\n \\int_{0}^{t} |E(t-s)F(z_{0,x}^{\\varphi}(s))|_{\\dot{H}^{2}} \\diff{s}\n +\n \\int_{0}^{t} |E(t-s)Q^{\\frac12}\\varphi(s)|_{\\dot{H}^{2}} \\diff{s}\n \\\\\\leq&~\n |x|_{\\dot{H}^{2}}\n +\n \\int_{0}^{t} |(-A)^{\\frac{1}{2}}E(t-s)|_{\\mathcal{L}(H)}\n |F(z_{0,x}^{\\varphi}(s))|_{\\dot{H}^{1}} \\diff{s}\n \\\\&~+\n \\int_{0}^{t} |(-A)E(t-s)Q^{\\frac12}|_{\\mathcal{L}(H)}|\\varphi(s)| \\diff{s}\n \\\\\\leq&~\n |x|_{\\dot{H}^{2}}\n +\n C(1+|x|_{\\dot{H}^{1}})\n +\n |\\varphi|_{L^{2}(0,T;H)}\n \\Big(\\int_{0}^{t} |(-A)E(t-s)Q^{\\frac12}|_{\\mathcal{L}_{2}(H)}^{2} \\diff{s}\\Big)^{\\frac{1}{2}}\n \\\\\\leq&~\n |x|_{\\dot{H}^{2}} + C(1+|x|_{\\dot{H}^{2}})\n +\n C|\\varphi|_{L^{2}(0,T;H)} |(-A)^{\\frac{1}{2}}Q^{\\frac12}|_{\\mathcal{L}_{2}(H)}\n \\\\\\leq&~\n C(1+|x|_{\\dot{H}^{2}}),\n \\end{align*}\n where we have used \\eqref{eq:auxiliaryassumptionFuH1} and \\cite[Lemma 2.3]{wang2015note}.\n Thus we complete the proof.\n\\end{proof}\n\n\n\n\nLemma \\ref{lem:spatialregularity} means that\nthe mild solution $\\{z_{0,x}^{\\varphi}(t)\\}_{t \\in [0,T]}$ is also a strong solution. To proceed, we denote by $W_{2}^{1,2}(T)$ the closure of $C^{\\infty}([0,T] \\times [0,1];\\mathbb{R})$ in the norm\n\\begin{equation*}\n\\begin{split}\n |z|_{W_{2}^{1,2}(T)}\n :=&\n \\Big(\n \\int_{0}^{T}\\int_{0}^{1}\n |z(t,\\xi)|^{2}\n +\n \\Big|\\frac{\\partial z(t,\\xi)}{\\partial t}\\Big|^{2}\n +\n \\Big|\\frac{\\partial z(t,\\xi)}{\\partial\\xi}\\Big|^{2}\n +\n \\Big|\\frac{\\partial^{2} z(t,\\xi)}{\\partial\\xi^{2}}\\Big|^{2}\n \\diff{\\xi}\\diff{t}\n \\Big)^{\\frac{1}{2}}.\n\\end{split}\n\\end{equation*}\nAs $Q$ is one-to-one, we have\n\\begin{equation}\\label{eq:I0Txz\n I_{0,T}^{x}(z)\n =\n \\begin{cases}\n \\frac{1}{2}\n \\int_{0}^{T}\n \\Big|Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n -\n Az(t) - F(z(t))\\Big)\n \\Big|^{2}\n \\diff{t}\n , & \\mbox{if } z \\in W_{2,Q}^{1,2}(T) \\text{~with~} z(0) = x, \\\\\n +\\infty, & \\text{otherwise},\n \\end{cases}\n\\end{equation}\nwhere\n\\begin{align*}\n W_{2,Q}^{1,2}(T)\n :=\n \\Big\\{&z \\in W_{2}^{1,2}(T):\n \\frac{\\diff{z(t)}}{\\diff{t}} - Az(t) - F(z(t)) \\in Q^{\\frac{1}{2}}(H) \\text{~for almost all~} t \\in [0,T]\n \\\\&\\int_{0}^{T}\n \\Big|Q^{-\\frac{1}{2}}\\Big(\\frac{\\diff{z(t)}}{\\diff{t}} - Az(t) - F(z(t))\\Big)\\Big|^{2}\\diff{t} < +\\infty\\Big\\}.\n\\end{align*}\n\n\n\n\nFor the rate function $I_{0,T}^{n,y}$, its effective domain is given by\n$$\n \\mathcal{H}_{1}^{n,y}(T)\n :=\n \\Big\\{z \\in C([0,T];H_{n}):\n z(\\cdot) = y + \\int_{0}^{\\cdot} \\upsilon(s) \\diff{s},\n \\upsilon \\in L^{2}(0,T;H_{n})\\Big\\}.\n$$\nIt follows that\n\\begin{equation}\\label{eq:I0Tnyz\n I_{0,T}^{n,y}(z)\n =\n \\begin{cases}\n \\frac{1}{2}\n \\int_{0}^{T}\n \\Big|Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n -\n A_{n}z(t) - F_{n}(z(t))\\Big)\n \\Big|^{2}\n \\diff{t}\n , & \\mbox{if~} z \\in \\mathcal{H}_{1}^{n,y}(T), \\\\\n +\\infty, & \\mbox{otherwise},\n \\end{cases}\n\\end{equation}\nwhere $Q_{n}^{-\\frac{1}{2}} \\colon H_{n} \\to H_{n}$ is the inverse operator of the bijective operator $Q_{n}^{\\frac{1}{2}} \\colon H_{n} \\to H_{n}$.\n\n\n\nSimilarly to \\cite[Definition 4.2]{chen2020large}, we introduce the definition of weakly asymptotical preservation for the LDP of sample paths by a numerical method.\n\\begin{Definition\n\\label{def:semiweakasympres\n \n We say that\n the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$, there exist $n \\in \\mathbb{N}^{+}$ and $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$ such that\n $$\n |z-z_{n}|_{C([0,T];H)} < \\kappa,\n \\quad\\quad\n |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})| < \\kappa.\n $$\n\\end{Definition}\n\n\n\nAfter these preparations, we have the following result.\n\n\\begin{Theorem}\n\\label{thm:spatialsolutionwap\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. If $x \\in \\dot{H}^{2}$, then\n \n the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\\begin{proof}\n For any $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$,\n \n we define\n $$\\varphi(t)\n :=\n Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n -\n Az(t) - F(z(t))\\Big)$$\n for almost all $t \\in [0,T]$. Then $\\varphi \\in L^2(0,T;H)$, $z_{0,x}^{\\varphi} = z$ and\n \\begin{equation}\\label{eq:IoTxz311311\n \\frac{1}{2}|\\varphi|_{L^2(0,T;H)}^{2} = I_{0,T}^{x}(z) < +\\infty,\n \\end{equation}\n which together with \\eqref{eq:39393939} in Lemma \\ref{lem:spatialregularity} yields\n \\begin{equation}\\label{eq:ztdotH2z0x313\n |z(t)|\n =\n |z_{0,x}^{\\varphi}(t)|\n \\leq\n C(1+|x|),\\quad t \\in [0,T].\n \\end{equation}\n Now for any $n \\in \\mathbb{N}^{+}$, we define $\\{z_{n}(t)\\}_{t \\in [0,T]}$ by\n \\begin{equation}\\label{eq:dzntdt312\n \\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n =\n A_{n}z_{n}(t) + F_{n}(z_{n}(t))\n + Q_{n}^{\\frac{1}{2}}P_{n}\\varphi(t),\n \\quad t \\in (0,T],\n \\quad z_{n}(0) = P_{n}x.\n \\end{equation}\n It follows from $\\langle Au,u \\rangle \\leq -\\lambda_{1}|u|^{2}, u \\in H$ and \\eqref{eq:globalLF} that\n \\begin{align*}\n &\\frac{1}{2}\\frac{\\diff{|z(t)-z_{n}(t)|^{2}}}{\\diff{t}}\n =\n \\Big\\langle \\frac{\\diff{(z(t)-z_{n}(t))}}{\\diff{t}},\n z(t)-z_{n}(t) \\Big\\rangle\n \\\\=&\n \\langle A(z(t)-z_{n}(t)),z(t)-z_{n}(t) \\rangle\n +\n \\langle (I-P_{n})F(z(t)),z(t)-z_{n}(t) \\rangle\n \\\\&+\n \\langle F_{n}(z(t))-F_{n}(z_{n}(t)),z(t)-z_{n}(t) \\rangle\n +\n \\langle (I-P_{n})Q^{\\frac{1}{2}}\\varphi(t),\n z(t)-z_{n}(t) \\rangle\n \\\\\\leq&\n -\\lambda_{1}|z(t)-z_{n}(t)|^{2}\n +\n |(I-P_{n})F(z(t))||z(t)-z_{n}(t)|\n \\\\&+\n L_{F}|z(t)-z_{n}(t)|^{2}\n +\n |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(t)|\n |z(t)-z_{n}(t)|\n \\\\\\leq&\n L_{F}|z(t)-z_{n}(t)|^{2}\n +\n \\frac{1}{2\\lambda_{1}}|(I-P_{n})F(z(t))|^{2}\n +\n \\frac{1}{2\\lambda_{1}}\n |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(t)|^{2},\n \\end{align*}\n where the weighted Young inequality is used in the last step. Integrating yields\n \\begin{align*}\n |z(t)&-z_{n}(t)|^{2}\n \\leq\n |x-P_{n}x|^{2}\n +\n 2L_{F}\\int_{0}^{t}|z(s)-z_{n}(s)|^{2}\\diff{s}\n \\\\&+\n \\frac{1}{\\lambda_{1}}\\int_{0}^{t}\n |(I-P_{n})F(z(s))|^{2}\\diff{s}\n +\n \\frac{1}{\\lambda_{1}}\\int_{0}^{t}\n |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(s)|^{2}\\diff{s}\n \\end{align*}\n and consequently\n \\begin{align*}\n |z&-z_{n}|_{C([0,t];H)}^{2}\n \\leq\n \\Pi_{n}\n +\n 2L_{F}\\int_{0}^{t}|z-z_{n}|_{C([0,s];H)}^{2}\\diff{s}\n \\end{align*}\n with\n \\begin{equation}\n \\Pi_{n}\n :=\n |(I-P_{n})x|^{2}\n +\n \\frac{1}{\\lambda_{1}}\\int_{0}^{T}\n |(I-P_{n})F(z(s))|^{2}\\diff{s}\n +\n \\frac{1}{\\lambda_{1}}\\int_{0}^{T}\n |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(s)|^{2}\\diff{s}.\n \\end{equation}\n The Gronwall inequality shows\n \\begin{equation}\\label{eq:zzn314\n |z-z_{n}|_{C([0,t];H)}^{2} \\leq \\Pi_{n}e^{2L_{F}t} \\leq \\Pi_{n}e^{2L_{F}T}, \\quad t \\in [0,T].\n \\end{equation}\n Applying \\eqref{eq:globalLF} and \\eqref{eq:ztdotH2z0x313} yields $$|(I-P_{n})F(z(t))|^{2} \\leq |F(z(t))|^{2} = |F(z_{0,x}^{\\varphi}(t))|^{2} \\leq C(1+|x|^{2}),\\quad t \\in [0,T].$$\n We use $\\lim\\limits_{n \\to \\infty}|(I-P_{n})F(z(t))|^{2} = 0, t \\in [0,T]$ and the bounded convergence theorem to get\n \\begin{equation*}\n \\lim_{n \\to \\infty}\n \\frac{1}{\\lambda_{1}}\\int_{0}^{T}\n |(I-P_{n})F(z(s))|^{2}\\diff{s}\n =\n 0.\n \\end{equation*}\n In the same way, one can validate $\\lim\\limits_{n \\to \\infty}\\Pi_{n} = 0$ and thus $\\lim\\limits_{n \\to \\infty}|z-z_{n}|_{C([0,T];H)} = 0$ by \\eqref{eq:zzn314}. It follows that for any $\\kappa > 0$, there exists $n_{1} \\in \\mathbb{N}^{+}$ such that\n \\begin{equation}\\label{eq:z-znC0TH\n |z-z_{n}|_{C([0,T];H)} < \\kappa,\\quad n > n_{1}.\n \\end{equation}\n By \\eqref{eq:dzntdt312} and \\eqref{eq:I0Tnyz}, we have $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$ and hence\n \\begin{equation*}\n I_{0,T}^{n,y}(z_{n})\n =\n \\frac{1}{2}\n \\int_{0}^{T}\n \\Big|Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n -\n A_{n}z_{n}(t) - F_{n}(z_{n}(t))\\Big)\n \\Big|^{2}\n \\diff{t}.\n \\end{equation*}\n It immediately follows from \\eqref{eq:dzntdt312} that\n \\begin{equation}\\label{eq:I0Tny317317\n I_{0,T}^{n,y}(z_{n})\n =\n \\frac{1}{2}|P_{n}\\varphi|_{L^2(0,T;H)}^{2}.\n \\end{equation}\n Applying \\eqref{eq:IoTxz311311}, \\eqref{eq:I0Tny317317} and the H\\\"{o}lder inequality leads to\n \\begin{align*}\n |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})|\n =&\n \\frac{1}{2}\\Big|\\int_{0}^{T}\n \\big\\langle (I+P_{n})\\varphi(t),\n (I-P_{n})\\varphi(t)\\big\\rangle \\diff{t}\\Big|\n \\\\\\leq&\n |\\varphi|_{L^2(0,T;H)}\n \\Big(\\int_{0}^{T}\n |(I-P_{n})\\varphi(t)|^{2} \\diff{t}\\Big)^{\\frac{1}{2}}.\n \\end{align*}\n By the Lebesgue dominated convergence theorem, we conclude\n \\begin{equation}\\label{eq:313313313\n \\lim\\limits_{n \\to \\infty}\n |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})| = 0,\n \\end{equation}\n which yields that there exists $n_{2} \\in \\mathbb{N}^{+}$ such that\n \\begin{equation}\\label{eq:I0Txz-I0Tnyzn\n |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})|\n <\n \\kappa, \\quad n > n_{2}.\n \\end{equation}\n By choosing $n > \\max\\{n_{1},n_{2}\\}$, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{LDP for invariant measures of spatial discretization}\n\\label{subsec:semiinvameaLDPs\nAs is well known, the uniform LDP of sample paths not only is of importance in itself but also can be used to derive the LDP of invariant measures; see, e.g., \\cite{sowers1992largeinvariant,cerrai2005largeinvariant}. With this purpose, one must first guarantee the existence of invariant measures of spatial discretization \\eqref{eq:SPDEsemi}.\nAccording to \\cite[Theorem 3.1]{CGW2020ErgodicityANM} and \\cite[Remark 7.2 and Proposition 7.10]{da2006introduction},\nthe family of probability measures $\\{\\mu_{t}^{\\varepsilon,n}\\}_{t > 0}$, defined by\n $$\n \\mu_{t}^{\\varepsilon,n}(B)\n :=\n \\frac{1}{t} \\int_{0}^{t}\n \\P\\big(X_{0}^{\\varepsilon,n}(s) \\in B\\big)\n \\diff{s},\n \\quad B \\in \\mathcal{B}(H_{n}), t > 0\n $$\nis tight. It follows from the Krylov--Bogoliubov theorem (\\cite[Theorem 7.1]{da2006introduction}) that there exists $\\{t_{i}\\}_{i \\in \\mathbb{N}^{+}} \\uparrow +\\infty$ (possibly depending on $\\varepsilon$) such that the sequence $\\{\\mu_{t_{i}}^{\\varepsilon,n}\\}_{i \\in \\mathbb{N}^{+}}$ converges weakly to some probability measure $\\mu^{\\varepsilon,n}$ on $(H_{n},\\mathcal{B}(H_{n}))$, which is invariant for \\eqref{eq:SPDEsemi}.\nTo study the LDP of invariant measures $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$, we define $V^{n} \\colon H_{n} \\to [0,+\\infty]$ by\n\\begin{equation}\\label{eq:Vny\n V^{n}(v)\n =\n \\inf\\{I_{0,T}^{n,0}(z) : T > 0, z \\in C([0,T];H_{n}), z(0) = 0, z(T) = v \\},\n\\end{equation}\nand its level set\n$K^{n}(\\alpha) = \\{v \\in H_{n} : V^{n}(v) \\leq \\alpha\\}$ for each $\\alpha \\geq 0$.\n\n\n\n\n\n\n\n\n\n\\subsubsection{$V^{n}$ being a good rate function}\n\n\n\n\n\n\nIn this part, we will use the boundedness of $K^{n}$ to show that $V^{n}$ is a good rate function. To this end, we need the following lemma.\n\n\n\\begin{Lemma}\\label{lem:4.9\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold and let $\\{z(t)\\}_{t \\in [0,T]}$ be the mild solution of the following equation\n \\begin{equation}\\label{eq:diffztdifft\n \\frac{\\diff{z(t)}}{\\diff{t}}\n =\n A_{n}z(t) + F_{n}(z(t)) + Q_{n}^{\\frac12}\\psi(t),\n \\quad t \\in (0,T],\n \\quad z(0) = y \\in H_{n}\n \\end{equation}\n with $\\psi \\in L^{2}(0,T;H_{n})$. Then we have\n \\begin{equation}\\label{eq:ztHn2\n |z(t)|^{2}\n \\leq\n C\\big(1+|y|^{2}\n +\n |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\n |\\psi|_{L^{2}(0,T;H)}^{2}\\big),\n \\quad t \\in [0,T].\n \\end{equation}\n\\end{Lemma}\n\n\n\n\n\\begin{proof}\n Setting $O(t) := \\int_{0}^{t} E_{n}(t-s) Q_{n}^{\\frac12}\\psi(s) \\diff{s}, t \\in [0,T]$, we use $|E_{n}(t)|_{\\mathcal{L}(H_{n})} \\leq e^{-\\lambda_{1}t},t \\in [0,T]$ and the H\\\"{o}lder inequality to get\n \\begin{equation}\\label{eq:OtHn\n \\begin{split}\n |O(t)|\n \\leq&\n |Q_{n}^{\\frac12}|_{\\mathcal{L}(H)}\n \\int_{0}^{t} e^{-\\lambda_{1}(t-s)}|\\psi(s)| \\diff{s}\n \\leq\n |Q^{\\frac12}|_{\\mathcal{L}(H)}\n |\\psi|_{L^{2}(0,T;H)},\n \\quad t \\in [0,T].\n \\end{split}\n \\end{equation}\n Let $\\bar{z}(t) := z(t)-O(t), t \\in [0,T]$, then $\\{\\bar{z}(t)\\}_{t \\in [0,T]}$ satisfies\n $$\n \\frac{\\dif{\\bar{z}(t)}}{\\dif{t}}\n =\n A_{n}\\bar{z}(t) + F_{n}(\\bar{z}(t)+O(t)),\n \\quad t \\in (0,T],\n \\quad \\bar{z}(0) = y.\n $$\n Using \\eqref{eq:dissipativecond}\n \n \n and \\eqref{eq:globalLF}\n \n yields\n \\begin{align*}\n &~\\frac{\\diff{e^{ct}|\\bar{z}(t)|^{2}}}\n {\\diff{t}}\n =\n ce^{ct}|\\bar{z}(t)|^{2}\n +\n 2e^{ct}\\big\\langle\n A_{n}\\bar{z}(t) + F_{n}(\\bar{z}(t)+O(t)),\\bar{z}(t)\n \\big\\rangle\n \\\\=&~\n ce^{ct}|\\bar{z}(t)|^{2}\n +\n 2e^{ct}\\big\\langle\n A_{n}\\bar{z}(t) + F_{n}(\\bar{z}(t)),\\bar{z}(t)\n \\big\\rangle\n +\n 2e^{ct}\\big\\langle\n F_{n}(\\bar{z}(t)+O(t))-F_{n}(\\bar{z}(t)),\\bar{z}(t)\n \\big\\rangle\n \\\\\\leq&~\n -\n ce^{ct}|\\bar{z}(t)|^{2}\n +\n 2L_{F}e^{ct}|O(t)||\\bar{z}(t)|\n \\leq\n \\frac{L_{F}^{2}}{c}e^{ct}|O(t)|^{2}\n \\end{align*}\n due to the weighted Young inequality $ab \\leq \\kappa a^{2} +\\frac{b^{2}}{4\\kappa}$ for all $a,b \\in \\mathbb{R}$ with $\\kappa = c > 0$ in the last step. Applying \\eqref{eq:OtHn} leads to\n \\begin{equation*}\n |\\bar{z}(t)|^{2}\n \\leq\n |y|^{2}\n +\n \\frac{L_{F}^{2}}{c^{2}}\n |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\n |\\psi|_{L^{2}(0,T;H)}^{2},\n \\quad t \\in [0,T].\n \\end{equation*}\n Together with $|z(t)|^{2} \\leq 2|\\bar{z}(t)|^{2} + 2|O(t)|^{2}$ and \\eqref{eq:OtHn}, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\begin{Theorem\n\\label{th:semiinvariantgood\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then the mapping $V^{n}$ given by \\eqref{eq:Vny} is a good rate function.\n \n\\end{Theorem}\n\n\n\n\n\\begin{proof}\n Notice that $V^{n}$ is lower semi-continuous and $H_{n}$ is a finite dimensional space. In order to verify $V^{n}$ being a good rate function, it suffices to show that $K^{n}(\\alpha)$ is bounded. For any $y \\in K^{n}(\\alpha)$, the definition of $V^{n}(y)$ implies that for any $\\kappa > 0$ there exists $T_{\\kappa} > 0, z_{\\kappa} \\in C([0,T_{\\kappa}];H_{n}), z_{\\kappa}(0) = 0, z_{\\kappa}(T_{\\kappa}) = y$ such that\n \\begin{equation*}\n I_{0,T_{\\kappa}}^{n,0}(z_{\\kappa})\n \\leq\n V^{n}(y) + \\kappa\n \\leq\n \\alpha + \\kappa.\n \\end{equation*}\n The definition of $I_{0,T_{\\kappa}}^{n,0}(z_{\\kappa})$ means that there exists $\\psi_{\\kappa} \\in L^{2}(0,T_{\\kappa};H_{n})$ with $z_{0,0}^{n,\\psi_{\\kappa}} = z_{\\kappa}$ such that\n \\begin{equation*}\n \\frac{1}{2}|\\psi_{\\kappa}|_{L^{2}(0,T_{\\kappa};H)}^{2}\n \\leq\n I_{0,T_{\\kappa}}^{n,0}(z_{\\kappa}) + \\kappa\n \\leq\n \\alpha + 2\\kappa.\n \\end{equation*}\n Because of $z_{0,0}^{n,\\psi_{\\kappa}} = z_{\\kappa}$ satisfying \\eqref{eq:diffztdifft}, we use Lemma \\ref{lem:4.9} to get\n \\begin{equation*}\n |y|^{2}\n =\n |z_{\\kappa}(T_{\\kappa})|^{2}\n \\leq\n C\\big(1\n +\n |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\n |\\psi_{\\kappa}|_{L^{2}(0,T_{\\kappa};H)}^{2}\\big)\n \\leq\n C\\big(1\n +\n 2(\\alpha+2\\kappa)\n |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\\big).\n \\end{equation*}\n Thus we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Lower bound estimate for the LDP of $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$}\n\n\n\n\n\n\n\\begin{Theorem\n\\label{th:semiinvariantmeasurelowerbounded}\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $n \\in \\mathbb{N}^{+}$, $\\bar{y} \\in H_{n}$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n \\begin{equation}\\label{eq:muvnyHndelta\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y-\\bar{y}| < \\delta\\})\n \\geq\n \\exp\\Big(-\\frac{V^{n}(\\bar{y})+\\gamma}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0}.\n \\end{equation}\n\\end{Theorem}\n\n\\begin{proof\n Without loss of generality, we assume $V^{n}(\\bar{y}) < +\\infty$, otherwise \\eqref{eq:muvnyHndelta} obviously holds. For any $\\bar{y} \\in H_{n}$ and $\\gamma > 0$, the definition of $V^{n}(\\bar{y})$ implies that there exists $\\bar{T} > 0$, $\\bar{z} \\in C([0,\\bar{T}];H_{n})$ with $\\bar{z}(0) = 0$ and $\\bar{z}(\\bar{T}) = \\bar{y}$ such that\n $\n I_{0,\\bar{T}}^{n,0}(\\bar{z}) \\leq V^{n}(\\bar{y}) + \\frac{\\gamma}{4}.\n $\n The definition of $I_{0,\\bar{T}}^{n,0}(\\bar{z})$\n further implies that there exists $\\bar{\\varphi} \\in L^{2}(0,\\bar{T};H_{n})$ with $z_{0,0}^{n,\\bar{\\varphi}} = \\bar{z}$ such that\n $$\n \\frac{1}{2}|\\bar{\\varphi}|_{L^{2}(0,\\bar{T};H)}^{2}\n \\leq\n I_{0,\\bar{T}}^{n,0}(\\bar{z}) + \\frac{\\gamma}{4}\n \\leq\n V^{n}(\\bar{y}) + \\frac{\\gamma}{2}.\n $$\n For any $T > \\bar{T}$, $K > 0$ and $y \\in H_{n}$ with $|y| \\leq K$, we define\n $$\n \\tilde{\\varphi}(t)\n :=\n \\left\\{\n \\begin{array}{ll}\n 0, & t \\in [0,T-\\bar{T}], \\\\\n \\bar{\\varphi}(t-(T-\\bar{T})), & t \\in [T-\\bar{T},T]\n \\end{array}\n \\right.\n $$\n and\n $$\n \\tilde{z}(t)\n :=\n z_{0,y}^{n,\\tilde{\\varphi}}(t)\n =\n \\left\\{\n \\begin{array}{ll}\n z_{0,y}^{n,0}(t), & t \\in [0,T-\\bar{T}], \\\\\n z_{T-\\bar{T},z_{0,y}^{n,0}(T-\\bar{T})}^{n,\\bar{\\varphi}(\\cdot-(T-\\bar{T}))}(t),\n & t \\in [T-\\bar{T},T].\n \\end{array}\n \\right.\n $$\n Then\n we have\n $|\\tilde{\\varphi}|_{L^{2}(0,T;H)}^{2}\n =\n |\\bar{\\varphi}|_{L^{2}(0,\\bar{T};H)}^{2}$\n ,\n $\\tilde{z} \\in C([0,T];H_{n})$ and\n \\begin{equation}\\label{eq:I0Tnytildez\n \\begin{split}\n I_{0,T}^{n,y}(\\tilde{z})\n =&\n I_{0,T}^{n,y}(z_{0,y}^{n,\\tilde{\\varphi}})\n =\n \\frac{1}{2}\\inf\\{|\\varphi|_{L^{2}(0,T;H)}^{2}:\n \\varphi \\in L^{2}(0,T;H_{n}), z_{0,y}^{n,\\varphi} = \\tilde{z}\\}\n \\\\\\leq&\n \\frac{1}{2}|\\tilde{\\varphi}|_{L^{2}(0,T;H)}^{2}\n =\n \\frac{1}{2}|\\bar{\\varphi}|_{L^{2}(0,\\bar{T};H)}^{2}\n \\leq\n V^{n}(\\bar{y}) + \\frac{\\gamma}{2}.\n \\end{split}\n \\end{equation}\n In view of\n \\begin{equation*}\n \\begin{split}\n \\tilde{z}(t)\n =&\n E_{n}(t-(T-\\bar{T}))z_{0,y}^{n,0}(T-\\bar{T})\n +\n \\int_{0}^{t-(T-\\bar{T})} E_{n}(t-(T-\\bar{T})-s)\n F_{n}(\\tilde{z}(s+T-\\bar{T})) \\diff{s}\n \\\\&+\n \\int_{0}^{t-(T-\\bar{T})} E_{n}(t-(T-\\bar{T})-s)\n Q_{n}^{\\frac12} \\bar{\\varphi}(s) \\diff{s},\n \\quad t \\in [T-\\bar{T},T],\n \\end{split}\n \\end{equation*}\n we set $\\hat{z}(t) := \\tilde{z}(t+(T-\\bar{T})), t \\in [0,\\bar{T}]$ to get\n \\begin{equation*}\n \\begin{split}\n \\hat{z}(t)\n =&\n E_{n}(t)z_{0,y}^{n,0}(T-\\bar{T})\n +\n \\int_{0}^{t} E_{n}(t-s) F_{n}(\\hat{z}(s)) \\diff{s}\n +\n \\int_{0}^{t} E_{n}(t-s)Q_{n}^{\\frac12}\\bar{\\varphi}(s) \\diff{s},\n \\quad t \\in [0,\\bar{T}]\n \\end{split}\n \\end{equation*}\n and consequently\n \\begin{equation*}\n \\hat{z}(t)-z_{0,0}^{n,\\bar{\\varphi}}(t)\n =\n E_{n}(t)z_{0,y}^{n,0}(T-\\bar{T})\n +\n \\int_{0}^{t} E_{n}(t-s)\n \\big( F_{n}(\\hat{z}(s)) - F_{n}(z_{0,0}^{n,\\bar{\\varphi}}(s)) \\big)\n \\diff{s},\n \\quad t \\in [0,\\bar{T}].\n \\end{equation*}\n Using $|E_{n}(t)|_{\\mathcal{L}(H_{n})} \\leq e^{-\\lambda_{1}t}$ for all $t \\geq 0$, \\eqref{eq:globalLF} and the Gronwall inequality yields\n \\begin{equation}\\label{eq:hatztz00nbar\n \\begin{split}\n |\\hat{z}(t)-z_{0,0}^{n,\\bar{\\varphi}}(t)|\n \\leq&\n |z_{0,y}^{n,0}(T-\\bar{T})| e^{L\\bar{T}},\n \\quad t \\in [0,\\bar{T}].\n \\end{split}\n \\end{equation}\n As $\\{z_{0,y}^{n,0}(t)\\}_{t \\in [0,T-\\bar{T}]}$ is the solution of\n \\begin{equation*}\n \\frac{\\diff{z_{0,y}^{n,0}(t)}}{\\diff{t}}\n =\n A_{n}z_{0,y}^{n,0}(t)\n +\n F_{n}(z_{0,y}^{n,0}(t)),\n \\quad t \\in (0,T-\\bar{T}],\n \\quad z_{0,y}^{n,0}(0) = y,\n \\end{equation*}\n we use \\eqref{eq:dissipativecond}\n \n to get\n \\begin{equation*}\n \\begin{split}\n \\frac{\\diff{e^{2ct}|z_{0,y}^{n,0}(t)|^{2}}}{\\diff{t}}\n =&\n 2ce^{2ct}|z_{0,y}^{n,0}(t)|^{2}\n +\n 2e^{2ct}\\big\\langle\n Az_{0,y}^{n,0}(t) + F(z_{0,y}^{n,0}(t)),\n z_{0,y}^{n,0}(t)\\big\\rangle\n \\leq\n 0,\n \\end{split}\n \\end{equation*}\n which yields $|z_{0,y}^{n,0}(t)| \\leq e^{-ct}|y|$ for all $t \\in [0,T-\\bar{T}]$ and thus\n $\\lim\\limits_{T \\to \\infty\n \\sup\\limits_{|y|_{H_{n}} \\leq K} |z_{0,y}^{n,0}(T-\\bar{T})| = 0$.\n Noting \\eqref{eq:hatztz00nbar}, $\\hat{z}(\\bar{T}) = \\tilde{z}(T) = z_{0,y}^{n,\\tilde{\\varphi}}(T)$ and $z_{0,0}^{n,\\bar{\\varphi}}(\\bar{T}) = \\bar{z}(\\bar{T}) = \\bar{y}$, there exists $\\tilde{T} > \\bar{T}$ such that\n $$\n \\sup_{|y| \\leq K}\n |z_{0,y}^{n,\\tilde{\\varphi}}(\\tilde{T})-\\bar{y}|\n \\leq\n \\frac{\\delta}{2}.\n $$\n This together with the invariance of $\\mu^{\\varepsilon,n}$ implies\n \\begin{equation*}\n \\begin{split}\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y-&\\bar{y}| < \\delta\\})\n =\n \\int_{H_{n}} \\P(|X_{y}^{\\varepsilon,n}(\\tilde{T})-\\bar{y}| < \\delta)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\geq&\n \\int_{{|y| \\leq K}} \\P(|X_{y}^{\\varepsilon,n}(\\tilde{T})-\\bar{y}| < \\delta)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\geq&\n \\int_{{|y| \\leq K}} \\P(|X_{y}^{\\varepsilon,n}(\\tilde{T})\n - z_{0,y}^{n,\\tilde{\\varphi}}(\\tilde{T})| < \\delta\/2)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\geq&\n \\int_{{|y| \\leq K}}\n \\P(|X_{y}^{\\varepsilon,n}\n -z_{0,y}^{n,\\tilde{\\varphi}}|_{C([0,\\tilde{T}];H)}\n < \\delta\/2)\n \\,\\mu^{\\varepsilon,n}(\\dif{y}).\n \\end{split}\n \\end{equation*}\n By \\eqref{eq:semisoluldplower} and \\eqref{eq:I0Tnytildez}, there exists $\\varepsilon_{1} > 0$ such that\n \\begin{equation}\\label{eq:muvnyHnybary\n \\begin{split}\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y-\\bar{y}| < \\delta\\})\n \\geq&~\n \\int_{{|y| \\leq K}}\n \\exp\\Big(-\\frac{I_{0,\\tilde{T}}^{n,y}(z_{0,y}^{n,\\tilde{\\varphi}})+\\gamma\/2}\n {\\varepsilon^{2}}\\Big)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\geq&~\n \\mu^{\\varepsilon,n}({|y| \\leq K})\n \\exp\\Big(-\\frac{V^{n}(\\bar{y})+\\gamma}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{1}.\n \\end{split}\n \\end{equation}\n It remains to estimate $\\mu^{\\varepsilon,n}({|y| \\leq K})$. Using the definition of $\\mu^{\\varepsilon,n}$ yields\n \\begin{equation}\\label{eq:muvarepsilonnyHnK\n \\mu^{\\varepsilon,n}(|y| > K)\n =\n \\lim_{i \\to \\infty}\n \\frac{1}{t_{i}} \\int_{0}^{t_{i}}\n \\P(|X_{0}^{\\varepsilon,n}(t)| > K) \\diff{t}.\n \\end{equation}\n Similarly to the proof of Lemma \\ref{lem:4.9}, we use \\eqref{eq:SPDEsemi} to obtain\n \\begin{equation*}\n |X_{0}^{\\varepsilon,n}(t)|\n \\leq\n \\varepsilon\\frac{L+c}{c}\n |\\Gamma^{n}|_{C([0,t];H)}\n \\end{equation*}\n with\n \\begin{equation}\\label{eq:Gammant\n \\Gamma^{n}(t)\n :=\n \\int_{0}^{t}\n E_{n}(t-s)Q_{n}^{\\frac12}\n \\diff{W_{n}(s)},\n \\quad t \\geq 0.\n \\end{equation}\n This together with the Chebyshev inequality shows that for any $t \\geq 0$ and $K > 0$,\n \\begin{equation}\\label{eq:PX0vntHnK\n \\begin{split}\n &\\P(|X_{0}^{\\varepsilon,n}(t)| > K)\n \\leq\n \\P\\Big(\n |\\Gamma^{n}|_{C([0,t];H)}\n >\n \\frac{cK}{\\varepsilon(L+c)}\\Big)\n \\leq\n \\varepsilon^{2}\\Big(\\frac{L+c}{cK}\\Big)^{2}C\n \\end{split}\n \\end{equation}\n with $C$ being independent of $t$, which is due to\n \\begin{equation}\\label{eq:GammanC0tHn2\n \\begin{split}\n \\mathbb{E}\\big[|\\Gamma^{n}|_{C([0,t];H)}^{2}\\big]\n =&~\n \\mathbb{E}\\Big[\\sup_{s \\in [0,t]}\n \\Big|\\int_{0}^{s}\n E_{n}(s-r)Q_{n}^{\\frac12}\n \\diff{W_{n}(r)}\n \\Big|^{2}\\Big]\n \\leq\n C\\int_{0}^{t}\n |E_{n}(t-r)Q_{n}^{\\frac12}\\big|_{\\mathcal{L}_{2}(H_{n})}^{2}\n \\diff{r}\n \\\\\\leq&~\n C|(-A_{n})^{-\\frac{1}{2}}|_{\\mathcal{L}(H_{n})}^{2}\n |(-A_{n})^{\\frac{1}{2}}Q_{n}^{\\frac12}|_{\\mathcal{L}_{2}(H_{n})}^{2}\n \\int_{0}^{t}\n |E_{n}(t-r)|_{\\mathcal{L}(H_{n})}^{2}\n \\diff{r}\n \\\\\\leq&~\n C\\int_{0}^{t}\n e^{-2\\lambda_{1}(t-r)}\n \\diff{r}\n \\leq\n C,\n \\quad t \\geq 0.\n \\end{split}\n \\end{equation}\n By setting $\\bar{K} > 0$, \\eqref{eq:muvarepsilonnyHnK} and \\eqref{eq:PX0vntHnK} lead to\n \\begin{equation*}\n \\lim_{\\varepsilon \\to 0}\n \\mu^{\\varepsilon,n}(|y| > \\bar{K})\n \\leq\n \\lim_{\\varepsilon \\to 0}\n \\sup_{t \\geq 0}\\P(|X_{0}^{\\varepsilon,n}(t)| > \\bar{K})\n \\leq\n 0\n \\end{equation*}\n and consequently\n $\n \\lim\\limits_{\\varepsilon \\to 0}\n \\mu^{\\varepsilon,n}({|y| \\leq \\bar{K}})\n =\n 1\n $.\n This in combination with \\eqref{eq:muvnyHnybary} shows that there exists sufficiently small $\\varepsilon_{0} \\leq \\varepsilon_{1}$ such that \\eqref{eq:muvnyHndelta} holds. Thus we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Upper bound estimate for the LDP of $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$}\n\nThe following lemma gives the exponential tail estimate for invariant measure $\\mu^{\\varepsilon,n}$ based on the Fernique theorem.\n\n\n\n\n\\begin{Lemma}\\label{lem:stochasticevolutiontailestimate}\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha \\geq 0$, there exists $\n \\bar{K} > 0$ such that\n \\begin{equation}\\label{eq:327327\n \\mu^{\\varepsilon,n}(\\{u \\in H_{n} : |u| > \\bar{K}\\})\n \\leq\n \\exp\\Big(-\\frac{\\alpha}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq 1.\n \\end{equation}\n\\end{Lemma}\n\n\n\n\n\\begin{proof}\n For any $K > 0$, it follows from the definition of $\\mu^{\\varepsilon,n}$ that\n \\begin{equation}\\label{eq:mvnHnyHnK\n \\mu^{\\varepsilon,n}(\\{u \\in H_{n} : |u| > K\\})\n =\n \\lim_{i \\to \\infty}\\frac{1}{t_{i}}\n \\int_{0}^{t_{i}} \\P(|X_{0}^{\\varepsilon,n}(t)| > K) \\diff{t}.\n \\end{equation}\n Similarly to the proof of Lemma \\ref{lem:4.9}, one can\n show that there exists $C > 0$ independent of $t$ such that\n $\n |X_{0}^{\\varepsilon,n}(t)|\n \\leq\n \\varepsilon C |\\Gamma^{n}(t)|,\n $\n where $\\Gamma^{n}(t)$ is defined by \\eqref{eq:Gammant}.\n Thus for any $K > 0$ and $\\kappa > 0$, we use the Markov inequality to get\n \\begin{equation}\\label{eq:PBigsup0leqsleqt\n \\begin{split}\n \\P(|X_{0}^{\\varepsilon,n}(t)| > K)\n \\leq&\n \\P\\Big(|\\Gamma^{n}(t)|\n > \\frac{K}{\\varepsilon C}\\Big)\n \\\\=&\n \\P\\Big(\n \\exp\\big(\\kappa|\\Gamma^{n}(t)|^{2}\\big)\n >\n \\exp\\Big(\\kappa\\Big(\\frac{K}{\\varepsilon C}\\Big)^{2}\\Big)\\Big)\n \\\\\\leq&\n \\exp\\Big(-\\kappa\\Big(\\frac{K}{\\varepsilon C}\\Big)^{2}\\Big)\n \\mathbb{E}\\big[\\exp\\big(\\kappa|\\Gamma^{n}(t)|^{2}\\big)\\big],\n \\quad t \\geq 0.\n \\end{split}\n \\end{equation}\n By \\cite[Proposition 4.28]{da2014stochastic},\n \\begin{equation*}\n \n \\Gamma^{n}(t)\n \\sim\n \\mathcal{N}\\Big(0,\\int_{0}^{t} E_{n}(t-r)Q_{n}E_{n}(t-r)\\diff{r}\\Big)\n \\end{equation*}\n and for any $i = 1,\\ldots,n$,\n \\begin{equation*}\n \\Big(\\int_{0}^{t} E_{n}(t-r)Q_{n}E_{n}(t-r)\\diff{r}\\Big)e_{i}\n =\n \\Big(\\int_{0}^{t} e^{-2\\lambda_{i}(t-r)}q_{i}\\diff{r}\\Big)e_{i}\n =\n \\frac{q_{i}}{2\\lambda_{i}}\n \\big(1 - e^{-2\\lambda_{i}t}\\big)e_{i}.\n \\end{equation*}\n Applying Fernique's theorem (\\cite[Proposition 1.13]{da2006introduction}) yields that for any $\\kappa < \\min\\big\\{\\frac{\\lambda_{i}}{q_{i}}:i =1, \\ldots,n\\big\\}$,\n \n \\begin{equation}\\label{eq:EBigexpBigkappa\n \\begin{split}\n &\n \\mathbb{E}\\Big[\n \\exp\\big(\\kappa|\\Gamma^{n}(t)|^{2}\\big)\n \\Big]\n =\n \\Big(\\prod_{i=1}^{n}\n \\Big(1 - \\kappa \\frac{q_{i}}{\\lambda_{i}}\n \\big(1 - e^{-2\\lambda_{i}t}\\big) \\Big)\\Big)^{-\\frac{1}{2}}\n \\leq\n \\Big(\\prod_{i=1}^{n}\n \\Big(1 - \\kappa \\frac{q_{i}}{\\lambda_{i}}\\Big)\\Big)^{-\\frac{1}{2}},\n \\quad t \\geq 0.\n \\end{split}\n \\end{equation}\n Combining \\eqref{eq:mvnHnyHnK}, \\eqref{eq:PBigsup0leqsleqt} and \\eqref{eq:EBigexpBigkappa} implies that one can choose\n \\begin{equation*}\n \\bar{K}\n \\geq\n C\\sqrt{\\frac{1}{\\kappa}\\Big(\\alpha\n + \\varepsilon^{2}\\ln \\Big(\\prod_{i=1}^{n}\n \\Big(1 - \\kappa \\frac{q_{i}}{\\lambda_{i}}\\Big)\\Big)^{-\\frac{1}{2}}\\Big)}\n \\end{equation*}\n such that \\eqref{eq:327327} holds. Thus we complete the proof.\n\\end{proof}\n\n\n\nFor any $T, L > 0$, we define\n\\begin{equation}\\label{eq:K0TnLalpha\n K_{0,T}^{n,L}(\\alpha)\n :=\n \\{z \\in C([0,T];H_{n}) :\n |z(0)| \\leq L,I_{0,T}^{n,z(0)}(z) \\leq \\alpha\\}.\n\\end{equation}\nSimilarly to the proof of \\cite[Lemma 7.1]{cerrai2005largeinvariant}, we have the following lemma.\n\n\n\n\\begin{Lemma}\\label{lem:zTzK0TnLalpha\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha \\geq 0$ and $\\delta > 0$, there exist $\\bar{L} > 0$ and $\\bar{T} > 0$ such that\n $$\n \\big\\{z(t): z \\in K_{0,t}^{n,\\bar{L}}(\\alpha)\\big\\}\n \\subset\n \\Big\\{y \\in H_{n}:\\rho^{H}(y,K^{n}(\\alpha)) < \\frac{\\delta}{2}\\Big\\},\n \\quad t \\geq \\bar{T},\n $$\n where $\\rho^{H}(u,U) := \\inf\\limits_{u' \\in U}|u-u'|, u \\in H, U \\subset H$.\n\\end{Lemma}\n\n\n\n\n\n\n\nFor any $K,L > 0$, $J \\in \\mathbb{N}^{+}$, we define\n\\begin{equation}\n H_{K,L,J\n :=\n \\{z \\in C([0,J];H_{n}) : |z(0)| \\leq K,\n |z(j)| > L, j = 1,2,\\ldots,J \\}.\n\\end{equation}\nThe proof of the following lemma is similar to that of Lemma 7.2 in \\cite{cerrai2005largeinvariant}.\n\n\\begin{Lemma}\\label{lem:infI0barJnyz310\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha,\\delta > 0$ and $K > 0$, there exists $\\bar{J} \\in \\mathbb{N}^{+}$ such that\n $$\n \\inf\\{I_{0,\\bar{J}}^{n,z(0)}(z) : z \\in H_{K,\\bar{L},\\bar{J}}\\}\n >\n \\alpha,\n $$\n where $\\bar{L}$ is the constant introduced in Lemma \\ref{lem:zTzK0TnLalpha} corresponding to $\\alpha$ and $\\delta$.\n\\end{Lemma}\n\n\n\n\n\n\n\nBased on the above lemmas, we are in a position to show the LDP upper bound estimate.\n\\begin{Theorem}\n\\label{th:semiinvariantmeasureupperbounded}\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha \\geq 0$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n $$\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n \\leq\n \\exp\\Big(-\\frac{\\alpha+\\gamma}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0}.\n $$\n\\end{Theorem}\n\n\\begin{proof\n By the invariance of $\\mu^{\\varepsilon,n}$ and \\eqref{eq:K0TnLalpha}, we have that for any $t \\geq 0$, $K,\\, L > 0$, $J \\in \\mathbb{N}^{+}$,\n \\begin{equation}\\label{eq:muvarepsilonnyHn\n \\begin{split}\n &\\mu^{\\varepsilon,n}(\\{y \\in H_{n} :\n \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n =\n \\int_{H_{n}} \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha))\n \\geq \\delta)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\leq&\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > K\\})\n +\n \\int_{|y| \\leq K}\n \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\leq&\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > K\\})\n +\n \\sup_{|y| \\leq K}\n \\P(X_{y}^{\\varepsilon,n} \\in H_{K,L,J})\n \\\\&+\n \\int_{|y| \\leq K}\n \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta,\n X_{y}^{\\varepsilon,n} \\notin H_{K,L,J})\n \\,\\mu^{\\varepsilon,n}(\\dif{y}).\n \\end{split}\n \\end{equation}\n For the last term in \\eqref{eq:muvarepsilonnyHn}, applying\n $\n \\{X_{y}^{\\varepsilon,n} \\in H_{K,L,J}\\}\n =\n \\bigcap\\limits_{j=1}^{J}\n \\{|X_{y}^{\\varepsilon,n}(j)| > L\\}\n $\n and the Markov property of $\\{X_{y}^{\\varepsilon,n}(t)\\}_{t \\geq 0}$ yields\n \\begin{align*}\\label{eq:intyHnleqK}\n &\\int_{|y| \\leq K}\n \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta,\n X_{y}^{\\varepsilon,n} \\notin H_{K,L,J})\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\leq&\n \\sum\\limits_{j=1}^{J}\n \\int_{|y| \\leq K}\n \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta,\n |X_{y}^{\\varepsilon,n}(j)| \\leq L\\big)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\=&\n \\sum\\limits_{j=1}^{J}\n \\int_{|y| \\leq K}\n \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta,\n |X_{y}^{\\varepsilon,n}(j-j)| \\leq L\\big)\n \\,\\mu^{\\varepsilon,n}(\\dif{y})\n \\\\\\leq&\n \\sum\\limits_{j=1}^{J}\n \\sup_{|y| \\leq L}\n \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta\\big),\n \\quad t > J.\n \\end{align*}\n Substituting the above estimate into \\eqref{eq:muvarepsilonnyHn} leads to\n \\begin{equation}\n \\begin{split}\n \\mu^{\\varepsilon,n}&(\\{y \\in H_{n} :\n \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n \\leq\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > K\\})\n \\\\&+\n \\sum\\limits_{j=1}^{J}\n \\sup_{|y| \\leq L}\n \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta\\big)\n +\n \\sup_{|y| \\leq K}\n \\P(X_{y}^{\\varepsilon,n} \\in H_{K,L,J}),\n \\quad t > J.\n \\end{split}\n \\end{equation}\n It follows from Lemma \\ref{lem:stochasticevolutiontailestimate} that for any $\\alpha \\geq 0$, there exists $\\bar{K} > 0$ such that\n \\begin{equation}\\label{eq:muvnyHnK\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > \\bar{K}\\})\n \\leq\n \\exp\\Big(-\\frac{\\alpha}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq 1.\n \\end{equation}\n Lemma \\ref{lem:zTzK0TnLalpha} shows that for any $\\alpha > 0$ and $\\delta > 0$, there exists $\\bar{L} > 0$ and $\\bar{T} > 0$ such that for any $t \\geq \\bar{T}$ and $y \\in H_{n}$ with $|y| \\leq \\bar{L}$,\n \\begin{equation*}\n \\rho^{H}(z(t),K^{n}(\\alpha)) < \\frac{\\delta}{2},\n \\quad z \\in K_{0,t}^{n,\\bar{L}}(\\alpha).\n \\end{equation*}\n Then\n the triangle inequality\n $\n \\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha))\n \\leq\n \\rho^{H}(X_{y}^{\\varepsilon,n}(t),z(t))\n +\n \\rho^{H}(z(t),K^{n}(\\alpha))\n $\n implies $$\\{\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta\\} \\subset\n \\{\\rho^{C([0,t];H)}(X_{y}^{\\varepsilon,n},z)\n \\geq \\frac{\\delta}{2}\\},\n \\quad z \\in K_{0,t}^{n,\\bar{L}}(\\alpha).$$\n By the arbitrariness of $z \\in K_{0,t}^{n,\\bar{L}}(\\alpha)$, we get\n $$\\{\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta\\} \\subset\n \\{\\rho^{C([0,t];H)}\n (X_{y}^{\\varepsilon,n},K_{0,t}^{n,y}(\\alpha))\n \\geq \\frac{\\delta}{2}\\},\n \\quad y \\in H_{n}, |y| \\leq \\bar{L}.$$\n This together with \\eqref{eq:semisoluldpupper} implies that for any $t \\geq \\bar{T}$ there exists $\\varepsilon(t) > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq \\bar{L}$,\n \\begin{align*}\n \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta\\big)\n \\leq\n \\P\\big(\\rho^{C([0,t];H)}(X_{y}^{\\varepsilon,n},\n K_{0,t}^{n,y}(\\alpha)) \\geq \\delta\/2\\big)\n \\leq\n \\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon(t).\n \\end{align*}\n Taking $t = \\bar{T} + J$ and $\\varepsilon_{1} := \\min\\{\\varepsilon(t-j), j = 1,\\ldots,J\\}$, we have\n $$\n \\sum\\limits_{j=1}^{J}\n \\sup_{|y| \\leq \\bar{L}}\n \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta\\big)\n \\leq\n J \\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{1}.\n $$\n Moreover, Lemma \\ref{lem:infI0barJnyz310} implies that there exists $\\bar{J} \\in \\mathbb{N}^{+}$ such that $\\rho^{C([0,\\bar{J}];H)}(H_{\\bar{K},\\bar{L},\\bar{J}},\n K_{0,\\bar{J}}^{n,y}(\\alpha)) > 0$. It follows from \\eqref{eq:semisoluldpupper} that there exists $\\varepsilon_{2} > 0$ such that\n \\begin{equation*}\n \\begin{split}\n \\sup_{|y| \\leq \\bar{K}}\n \\P(X_{y}^{\\varepsilon,n} \\in H_{\\bar{K},\\bar{L},\\bar{J}})\n \\leq&\n \\sup_{|y| \\leq \\bar{K}}\n \\P\\big(\\rho^{C([0,\\bar{J}];H)}\n (X_{y}^{\\varepsilon,n},K_{0,\\bar{J}}^{n,y}(\\alpha)) \\geq\n \\rho^{C([0,\\bar{J}];H)}(H_{\\bar{K},\\bar{L},\\bar{J}},\n K_{0,\\bar{J}}^{n,y}(\\alpha))\\big)\n \\\\\\leq&\n \\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{2}.\n \\end{split}\n \\end{equation*}\n Combining the above estimates shows that for any $\\varepsilon \\leq \\varepsilon_{3} := \\min\\{1,\\varepsilon_{1},\\varepsilon_{2}\\}$,\n $$\n \\mu^{\\varepsilon,n}(\\{y \\in H_{n} :\n \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n \\leq\n \\exp\\Big(-\\frac{\\alpha}{\\varepsilon^{2}}\\Big)\n +\n (1+\\bar{J})\\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big).\n $$\n By taking sufficiently small $\\varepsilon_{0} \\leq \\varepsilon_{3}$, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsection{Weakly asymptotical preservation for the LDP of $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$}\n\\label{sebsec:semiinvameaLDPasy\n\n\nTo estimate the error between the rate functions $V^{n}$ and $V$, we denote the effective domain of $V$ by\n\\begin{equation}\n \\mathcal{D}_{V}\n :=\n \\{u \\in H : V(u) < \\infty\\}.\n\\end{equation}\nNow we give the following definition of weakly asymptotical preservation for the LDP of invariant measures by a numerical method.\n\n\\begin{Definition\n\\label{def:semiinvaweakasympres\n\n \n We say that \n \n the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $u \\in \\mathcal{D}_{V}$, there exist $n \\in \\mathbb{N}^{+}$ and $u_{n} \\in H_{n}$ such that\n $$\n |u-u_{n}| < \\kappa,\n \\qquad\n |V(u)-V^{n}(u_{n})| < \\kappa.\n $$\n\\end{Definition}\n\n\n\n\\begin{Theorem}\\label{th:semiinvameaLDPasympre}\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. If\n \n \n $F \\equiv \\textbf{0}$,\n then \n \n the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\n\\begin{proof}\n For any $u \\in \\mathcal{D}_{V}$, we set $u_{n} := P_{n}u, n \\in \\mathbb{N}^{+}$ and obtain $\\lim\\limits_{n \\to \\infty}|u-u_{n}| = 0$,\n i.e., for each $\\kappa > 0$ there exists $n_{1} \\in \\mathbb{N}^{+}$ such that\n \\begin{equation}\\label{eq:xynHkappa\n |u-u_{n}| < \\kappa,\\quad n \\geq n_{1}.\n \\end{equation}\n The definition of $V(u)$ implies that for the above given $\\kappa > 0$ there exists $T_{\\kappa} > 0$ and $z_{\\kappa} \\in C([0,T_{\\kappa}];H)$ with $z_{\\kappa}(0) = 0, z_{\\kappa}(T_{\\kappa}) = u$ such that\n $$\n I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n \\leq\n V(u) + \\frac{\\kappa}{2} < +\\infty,\n $$\n which in combination with \\eqref{eq:I0Txz} yields $z_{\\kappa} \\in W_{2,Q}^{1,2}(T_{\\kappa})$. Define\n \\begin{equation*}\n \\varphi_{\\kappa}(t)\n =\n Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n \\end{equation*}\n for almost all $t \\in [0,T_{\\kappa}]$, then $\\varphi_{\\kappa} \\in L^{2}(0,T_{\\kappa};H), z_{0,0}^{\\varphi_{\\kappa}} = z_{\\kappa}$ and $\\frac{1}{2}|\\varphi_{\\kappa}|_{L^{2}(0,T_{\\kappa};H)}^{2} = I_{0,T_{\\kappa}}^{0}(z_{\\kappa}) < +\\infty$, which together with Lemma \\ref{lem:spatialregularity} gives\n \\begin{equation}\\label{eq:zkappatdotH2\n |z_{\\kappa}(t)|_{\\dot{H}^{2}}\n =\n |z_{0,0}^{\\varphi_{\\kappa}}(t)|_{\\dot{H}^{2}}\n \\leq\n C,\n \\quad t \\in [0,T_{\\kappa}].\n \\end{equation}\n As $P_{n}z_{\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{\\kappa}),n \\in \\mathbb{N}^{+}$, we assert that\n \\begin{equation}\\label{eq:I0Tkappan0Pnzkappa342}\n \n \\lim\\limits_{n \\to \\infty}\n I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})\n =\n I_{0,T_{\\kappa}}^{0}(z_{\\kappa}),\n \\end{equation}\n which shows that there exists $n_{2} \\in \\mathbb{N}^{+}$ such that\n \\begin{equation}\\label{eq:I0Tkappan0Pnzkappa343}\n \n |I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n -I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})|\n \\leq\n \\frac{\\kappa}{2},\n \\quad n \\geq n_{2}.\n \\end{equation}\n In fact, applying the identity\n $|u|^{2} - |v|^{2} + |u-v|^{2} = 2\\langle u,u-v \\rangle$ for all $u,v \\in H$, \\eqref{eq:I0Txz}, \\eqref{eq:I0Tnyz}\n and the H\\\"{o}lder inequality leads to\n \\begin{align*}\n &~|I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n -I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})|\n \\leq\n \\frac{1}{2}\n \\int_{0}^{T_{\\kappa}}\n \\Bigg|\n \\Big|\n Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n \\Big|^{2}\n \\\\&~-\n \\Big|\n Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n - A_{n}P_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big)\n \\Big|^{2}\n \\Bigg|\n \\diff{t}\n \\\\=&~\n \\frac{1}{2}\n \\int_{0}^{T_{\\kappa}}\n \\Bigg|\n -\\Big|Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n -\n Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n - AP_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big)\n \\Big|^{2}\n \\\\&~+\n 2\\Big\\langle\n Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n -\n Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n - AP_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big),\n \\\\&~~~~~~~~~~Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n \\Big\\rangle\n \\Bigg|\\diff{t}\n \\\\\\leq&~\n \\frac{1}{2}\\int_{0}^{T_{\\kappa}} \\Theta_{n}(t) \\diff{t}\n +\n \\Big(2I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n \\int_{0}^{T_{\\kappa}}\\Theta_{n}(t)\\diff{t}\\Big)^{\\frac{1}{2}},\n \\end{align*}\n where\n \\begin{equation*}\n \\Theta_{n}(t)\n :=\n \\Big|Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n -\n Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n - AP_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big)\n \\Big|^{2}.\n \\end{equation*}\n \n Observing $Q^{-\\frac12}u = Q_{n}^{-\\frac12}u,u \\in H_{n}$, we apply the mean value formula to get\n \n \\begin{align*}\n \\Theta_{n}(t)\n =&~\n \\Big|(I-P_{n})Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n -\n Q^{-\\frac{1}{2}}\n \\big(F_{n}(z_{\\kappa}(t)) - F_{n}(P_{n}z_{\\kappa}(t))\\big)\n \\Big|^{2}\n \\\\=&~\n \\Big|(I-P_{n})Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n \\\\&~-\n \\int_{0}^{1}Q^{-\\frac{1}{2}}\n F_{n}^{'}\\big(P_{n}z_{\\kappa}(t)\n +r(z_{\\kappa}(t)-P_{n}z_{\\kappa}(t))\\big)\n \\big(z_{\\kappa}(t)-P_{n}z_{\\kappa}(t)\\big)\\diff{r}\\Big|^{2}.\n \\end{align*}\n On one hand, we use \\eqref{eq:hybridAQ}, \\eqref{eq:auxiliaryassumption}, $F_{n}^{'} = P_{n}F'$ and $|I-P_{n}|_{\\mathcal{L}(H)} = 1, n \\in \\mathbb{N}^{+}$ to show\n \\begin{equation*}\n \\begin{split}\n \\Theta_{n}(t)\n \\leq\n 2\\Big|Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\\Big|^{2}\n +\n 2L^{2}\\big(1+4|z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\\big)^{2}\n |z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\n \\end{split}\n \\end{equation*}\n with\n \\begin{equation*}\n \\int_{0}^{T_{\\kappa}} \\Big|Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}} - Az_{\\kappa}(t)\n - F(z_{\\kappa}(t))\\Big)\\Big|^{2} \\diff{t}\n =\n 2I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n <\n \\infty,\n \\end{equation*}\n \\begin{equation*}\n \\int_{0}^{T_{\\kappa}}\n \\big(1+4|z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\\big)^{2}\n |z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\n \\diff{t}\n \\leq\n T_{\\kappa}C^{2}\\big(1+4C^{2}\\big)^{2}\n <\n \\infty\n \\end{equation*}\n due to \\eqref{eq:zkappatdotH2}. On the other hand, using $\\lim\\limits_{n \\to \\infty} P_{n}u = u, u \\in H$ and\n \\eqref{eq:auxiliaryassumption} yields\n \\begin{equation*}\n \\begin{split}\n \\lim\\limits_{n \\to \\infty} \\Theta_{n}(t)\n \\leq&\n 2\\lim\\limits_{n \\to \\infty}\n \\Big|(I-P_{n})Q^{-\\frac{1}{2}}\n \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\\Big|^{2}\n \\\\&+\n 2L^{2}\\big(1+4|(-A)z_{\\kappa}(t)|^{2}\\big)^{2}\n \\lim\\limits_{n \\to \\infty}\n |(I-P_{n})(-A)z_{\\kappa}(t)|^{2} = 0.\n \\end{split}\n \\end{equation*}\n Combining the above results and using the dominated convergence theorem ensure \\eqref{eq:I0Tkappan0Pnzkappa342}.\n As $P_{n}z_{\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{\\kappa}) \\subset C([0,T_{\\kappa}];H_{n})$ satisfying $P_{n}z_{\\kappa}(0) = 0$ and $P_{n}z_{\\kappa}(T_{\\kappa}) = P_{n}u = u_{n}$ for each $n \\in \\mathbb{N}^{+}$, the definition of $V^{n}(u_{n})$ and \\eqref{eq:I0Tkappan0Pnzkappa343} lead to\n \\begin{equation}\\label{eq:345\n V^{n}(u_{n})\n \\leq\n I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})\n \\leq\n I_{0,T_{\\kappa}}^{0}(z_{\\kappa}) + \\frac{\\kappa}{2}\n \\leq\n V(u) + \\kappa,\n \\quad n \\geq \\max\\{n_{1},n_{2}\\}.\n \\end{equation}\n It remains to show $V(u) \\leq V^{n}(u_{n}) + \\kappa$ for sufficiently large $n \\in \\mathbb{N}^{+}$.\n Now for each $n \\geq \\max\\{n_{1},n_{2}\\}$, the definition of $V^{n}(u_{n})$ implies that for the above given $\\kappa > 0$ there exists $T_{n,\\kappa} > 0$, $z_{n,\\kappa} \\in C([0,T_{n,\\kappa}];H_{n})$, $z_{n,\\kappa}(0) = 0$, $z_{n,\\kappa}(T_{n,\\kappa}) = u_{n}$ such that\n \\begin{equation}\\label{eq:346\n I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n \\leq\n V^{n}(u_{n}) + \\frac{\\kappa}{2}\n \\leq\n V(u) + \\frac{3}{2}\\kappa\n <\n + \\infty,\n \\end{equation}\n which in combination with \\eqref{eq:I0Tnyz} yields $z_{n,\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{n,\\kappa})$.\n Since\n $z_{n,\\kappa} \\in W_{2,Q}^{1,2}(T_{n,\\kappa})$, the definition of $V(u_{n})$ in \\eqref{eq:Vny} shows\n \\begin{equation}\\label{eq:347\n V(u_{n})\n \\leq\n I_{0,T_{n,\\kappa}}^{0}(z_{n,\\kappa}).\n \\end{equation}\n Due to\n $F \\equiv \\textbf{0}$,\n \n we have\n \\begin{equation}\\label{eq:348\n \\begin{split}\n I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n =&\n \\frac{1}{2}\n \\int_{0}^{T_{n,\\kappa}}\n \\Big|Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{z_{n,\\kappa}(t)}}{\\diff{t}}\n -\n A_{n}z_{n,\\kappa}(t)\\Big)\n \\Big|^{2}\n \\diff{t}\n \\\\=&\n \\frac{1}{2}\n \\int_{0}^{T_{n,\\kappa}}\n \\Big|Q^{-\\frac12}\n \\Big(\\frac{\\diff{z_{n,\\kappa}(t)}}{\\diff{t}}\n -\n Az_{n,\\kappa}(t)\\Big)\n \\Big|^{2}\n \\diff{t}\n =\n I_{0,T_{n,\\kappa}}^{0}(z_{n,\\kappa}).\n \\end{split}\n \\end{equation}\n As $\\lim\\limits_{n \\to \\infty} u_{n} = u$, the semi-lower continuity of $V$ implies $V(u) \\leq \\liminf\\limits_{n \\to \\infty} V(u_{n})$, which means that for the above given $\\kappa > 0$ there exists $n_{3} \\in \\mathbb{N}^{+}$ such that\n \\begin{equation}\\label{eq:349\n V(u) \\leq \\liminf_{n \\to \\infty} V(u_{n})\n \\leq\n V(u_{n}) + \\frac{\\kappa}{2},\n \\quad n \\geq n_{3}.\n \\end{equation}\n Combining \\eqref{eq:346}--\\eqref{eq:349} leads to\n \\begin{equation}\\label{eq:350\n V(u)\n \\leq\n V^{n}(u_{n}) + \\kappa,\n \\quad n \\geq \\max\\{n_{1},n_{2},n_{3}\\}.\n \\end{equation}\n Finally, \\eqref{eq:xynHkappa}, \\eqref{eq:345} and \\eqref{eq:350} finish the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Spatio-temporal discretization and its LDPs}\n\\label{sec:temporal\nIn Subsection \\ref{eq:AEEM}, we obtain a full-discrete numerical approximation $\\{Y_{y,m}^{\\varepsilon,n}\\}_{m \\in \\mathbb{N}}$ by applying the accelerated exponential Euler method to \\eqref{eq:SPDEsemi} and give the uniform LDP of sample paths of $\\{Y_{y}^{\\varepsilon, n}\\}_{\\varepsilon > 0}$, which is a continuous version of $\\{Y_{y,m}^{\\varepsilon,n}\\}_{m \\in \\mathbb{N}}$.\nThen the weakly asymptotical preservation for the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ is given in Theorem \\ref{th:fullLDPasympre}.\nSubsection \\ref{subsec:fullinvameaLDP} shows the LDP of invariant measures $\\{\\mu^{\\varepsilon,n,\\tau}\\}_{\\varepsilon > 0}$ of $\\{Y_{y}^{\\varepsilon, n}\\}_{\\varepsilon > 0}$ and establishes its weakly asymptotical preservation for the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\subsection{Accelerated exponential Euler method}\n\\label{eq:AEEM\n\n\n\n\nLet $\\tau > 0$ be a uniform time stepsize and $t_{m} := m\\tau, m \\in \\mathbb{N}$ the grid points, the accelerated exponential Euler method (see \\cite{jentzen2009overcoming}) applied to \\eqref{eq:SPDEsemi} is given\nby setting $Y_{y,0}^{\\varepsilon,n} = y$ and\n\\begin{equation}\\label{eq:EE\n Y_{y,m+1}^{\\varepsilon,n}\n =\n E_{n}(\\tau)Y_{y,m}^{\\varepsilon,n}\n +\n \\int_{t_{m}}^{t_{m+1}}\n E_{n}(t_{m+1}-s)F_{n}(Y_{y,m}^{\\varepsilon,n}) \\diff{s}\n +\n \\varepsilon \\int_{t_{m}}^{t_{m+1}} E_{n}(t_{m+1}-s) Q_{n}^{\\frac12}\\diff{W_{n}(s)}\n\\end{equation}\nfor each $m \\in \\mathbb{N}$. To give a continuous time approximation of $\\{Y_{y,m}^{\\varepsilon,n}\\}_{m \\in \\mathbb{N}}$, we define $\\{Y_{y}^{\\varepsilon,n}(t)\\}_{t \\geq 0}$ by\n\\begin{equation}\\label{eq:4242\n Y_{y}^{\\varepsilon,n}(t)\n =\n E_{n}(t-t_{m}) Y_{y,m}^{\\varepsilon,n}\n +\n \\int_{t_{m}}^{t} E_{n}(t-s)\n F_{n}(Y_{y,m}^{\\varepsilon,n}) \\diff{s}\n +\n \\varepsilon \\int_{t_{m}}^{t} E_{n}(t-s) Q_{n}^{\\frac12}\\diff{W_{n}(s)}\n\\end{equation}\nfor all $t \\in [t_{m},t_{m+1}), m \\in \\mathbb{N}$.\nLet $\\lfloor \\cdot \\rfloor$ represent the largest integer no larger than the corresponding constant.\nAs $Y_{y}^{\\varepsilon,n}(t_{m}) = Y_{y,m}^{\\varepsilon,n},m \\in \\mathbb{N}$, i.e., $Y_{y}^{\\varepsilon,n}(\\tau \\lfloor t\/\\tau \\rfloor) = Y_{y,\\lfloor t\/\\tau \\rfloor}^{\\varepsilon,n}, t \\geq 0$, we have\n\\begin{equation}\\label{eq:saptiotemporalconteq\n\\begin{split}\n &\\diff{Y_{y}^{\\varepsilon,n}(t)}\n =\n \\big(A_{n}Y_{y}^{\\varepsilon,n}(t)\n +\n F_{n}(Y_{y}^{\\varepsilon,n}(\\tau \\lfloor t\/\\tau \\rfloor)) \\big)\\diff{t}\n +\n \\varepsilon Q_{n}^{\\frac12}\\diff{W_{n}(t)},\n \\quad t > 0,\n \\quad Y_{y}^{\\varepsilon,n}(0) = y.\n\\end{split}\n\\end{equation}\nFor any $\\phi \\in L^{2}(0,T;H_{n})$, it follows from \\cite[Theorem 1.1]{peszat1994large} that the skeleton equation\n\\begin{equation*\n \\frac{\\diff{z_{0,y}^{n,\\tau,\\phi}(t)}}{\\diff{t}}\n =\n A_{n}z_{0,y}^{n,\\tau,\\phi}(t)\n +\n F_{n}(z_{0,y}^{n,\\tau,\\phi}(\\tau \\lfloor t\/\\tau \\rfloor))\n +\n Q_{n}^{\\frac12}\\phi(t),\n \\quad t > 0,\n \\quad z_{0,y}^{n,\\tau,\\phi}(0) = y\n\\end{equation*}\nadmits a unique mild solution $\\{z_{0,y}^{n,\\tau,\\phi}(t)\\}_{t \\geq 0}$.\nWe will prove that $\\{Y_{y}^{\\varepsilon, n}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H_{n})$ with the rate function\n$I_{0,T}^{n,\\tau,y} \\colon C([0,T];H_{n}) \\to [0,+\\infty]$\ndefined by\n\\begin{equation}\\label{eq:SPDEfullratefun\n I_{0,T}^{n,\\tau,y}(z)\n =\n \\frac{1}{2} \\inf\\{|\\phi|_{L^{2}(0,T;H)}^{2}:\n \\phi \\in L^{2}(0,T;H_{n}), z_{0,y}^{n,\\tau,\\phi}= z\\},\n \\quad z \\in C([0,T];H_{n}).\n\\end{equation}\nBy the arguments for \\eqref{eq:I0Tnyz}, we have\n\\begin{equation}\\label{eq:I0TnMyz\n I_{0,T}^{n,\\tau,y}(z)\n =\n \\begin{cases}\n \\frac{1}{2}\n \\int_{0}^{T}\n \\Big|Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n -\n A_{n}z(t) - F_{n}(z(\\tau \\lfloor t\/\\tau \\rfloor))\\Big)\n \\Big|^{2}\n \\diff{t}\n , & \\mbox{if } z \\in \\mathcal{H}_{1}^{n,y}(T), \\\\\n +\\infty, & \\mbox{otherwise}.\n \\end{cases}\n\\end{equation}\n\n\nSimilarly to the proof of Theorem \\ref{thm:semisolutionLDP}, we establish the uniform LDP of sample paths of\n$\\{Y_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$.\n\n\\begin{Theorem}\n\\label{th:fullLDPsamplepath\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then $\\{Y_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H_{n})$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $I_{0,T}^{n,\\tau,y}$ given by \\eqref{eq:SPDEfullratefun}, i.e.,\n \\begin{enumerate}\n \\item [(i)] compact level set: for any $T > 0$, $n \\in \\mathbb{N}^{+}$ and $y \\in H_{n}$, the level set $K_{0,T}^{n,\\tau,y}(\\alpha) :=\n \\{z \\in C([0,T];H_{n}) : I_{0,T}^{n,\\tau,y}(z) \\leq \\alpha\\}$\n is compact for every $\\alpha \\geq 0$;\n\n \\item [(ii)] uniform lower bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_{0} > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$ and $z \\in K_{0,T}^{n,\\tau,y}(\\alpha)$,\n \\begin{equation}\\label{eq:fullsoluldplower\n \\P\\big(|Y_{y}^{\\varepsilon,n}-z|_{C([0,T];H)} < \\delta\\big)\n \\geq\n \\exp\\Big(-\\frac{I_{0,T}^{n,\\tau,y}(z)+\\gamma}{\\varepsilon^2}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0};\n \\end{equation}\n \\item [(iii)] uniform upper bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_0 > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$,\n \\begin{equation}\n \\P\\big(\\rho^{C([0,T];H)}(Y_{y}^{\\varepsilon,n},\n K_{0,T}^{n,\\tau,y}(\\alpha)) \\geq \\delta\\big)\n \\leq\n \\exp\\Big(-\\frac{\\alpha-\\gamma}{\\varepsilon^2}\\Big),\n \\quad \\varepsilon \\leq \\varepsilon_{0}.\n \\end{equation}\n \\end{enumerate}\n\\end{Theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo characterize the error between the rate functions $I_{0,T}^{n,\\tau,y}$ and $I_{0,T}^{x}$, we give the definition of weakly asymptotical preservation for the LDP of sample paths by a numerical method.\n\\begin{Definition}\n \n We say that\n the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$, there exist $n \\in \\mathbb{N}^{+}$, $\\tau > 0$ and $z_{n,\\tau} \\in \\mathcal{H}_{1}^{n,y}(T)$ such that\n $$\n |z-z_{n,\\tau}|_{C([0,T];H)} < \\kappa,\n \\quad\\quad\n |I_{0,T}^{x}(z)-I_{0,T}^{n,\\tau,y}(z_{n,\\tau})| < \\kappa.\n $$\n\\end{Definition}\n\n\nNow we show that $\\{Y_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\begin{Theorem}\n\\label{th:fullLDPasympre\n Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then\n the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\n\\begin{proof}\n For any $\\kappa > 0$ and $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$, it follows from Theorem \\ref{thm:spatialsolutionwap} that there exist $n \\in \\mathbb{N}^{+}$ and $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$ such that\n \\begin{equation}\\label{eq:part111\n |z-z_{n}|_{C([0,T];H)} < \\frac{\\kappa}{2} < \\kappa,\n \\quad\\quad\n |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})| < \\frac{\\kappa}{2}.\n \\end{equation}\n Then we use \\eqref{eq:I0Tnyz}, \\eqref{eq:I0TnMyz}, the identity\n $|u|^{2} - |v|^{2} + |u-v|^{2} = 2\\langle u,u-v \\rangle, u,v \\in H$ and the H\\\"{o}lder inequality to get\n \\begin{align*\n |I_{0,T}^{n,y}(z_{n})\n -I_{0,T}^{n,\\tau,y}(z_{n})|\n \\leq&~\n \\frac{1}{2}\n \\int_{0}^{T}\n \\Bigg|\n \\Big|Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n -\n A_{n}z_{n}(t)\n -\n F_{n}(z_{n}(t))\\Big)\\Big|^{2}\n \\\\&~-\n \\Big|Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n -\n A_{n}z_{n}(t)\n -\n F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\\Big)\n \\Big|^{2}\n \\Bigg|\n \\diff{t}\n \\\\=&~\n \\frac{1}{2}\n \\int_{0}^{T}\n \\Big|\n 2\\Big\\langle\n Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n - F_{n}(z_{n}(t))\\big),\n \\\\&~Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n -\n A_{n}z_{n}(t) - F_{n}(z_{n}(t))\\Big)\\Big\\rangle\n \\\\&~-\n \\big|Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n -F_{n}(z_{n}(t))\\big)\\big|^{2}\n \\Big|\n \\diff{t}\n \\\\\\leq&~\n \\Big(2I_{0,T}^{n,y}(z_{n})\\int_{0}^{T}\n \\big|Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n -\n F_{n}(z_{n}(t))\\big)\n \\big|^{2}\\diff{t}\\Big)^{\\frac12}\n \\\\&~+\n \\frac{1}{2}\\int_{0}^{T}\n \\big|Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n -\n F_{n}(z_{n}(t))\\big)\\big|^{2}\\diff{t}.\n \\end{align*}\n Applying \\eqref{eq:hybridAQ}, \\eqref{eq:auxiliaryassumption} and the mean value formula,\n \n we have\n \\begin{align*}\n &~\\big|Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n -\n F_{n}(z_{n}(t))\\big)\\big|^{2}\n \\\\\\leq&~\n \\big|Q_{n}^{-\\frac12}(-A_{n})^{-1}\\big|_{\\mathcal{L}(H_{n})}\n \\int_{0}^{1}\\big|(-A_{n})F_{n}^{'}\n \\big(z_{n}(t)\n +r(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n -z_{n}(t))\\big)\n \\big(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n -z_{n}(t)\\big)\n \\big|^{2}\\diff{r}\n \\\\\\leq&~\n C\\big(1+|z_{n}(t)|_{\\dot{H}^{2}}^{2}\n +|z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)|_{\\dot{H}^{2}}^{2}\\big)^{2}\n \\big|(-A_{n})\\big(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n -z_{n}(t)\\big)\\big|^{2},\n \\end{align*}\n where $F_{n}^{'} = P_{n}F'$. As $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$, we define\n $$\n \\psi(t)\n :=\n Q_{n}^{-\\frac12}\n \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n -\n A_{n}z_{n}(t)\n -\n F_{n}(z_{n}(t))\\Big)\n $$\n for almost all $t \\in [0,T]$. Then $\\psi \\in L^{2}(0,T;H_{n})$, $z_{0,y}^{n,\\psi} = z_{n}$ and $\\frac{1}{2}|\\psi|_{L^{2}(0,T;H)}^{2} = I_{0,T}^{n,y}(z_{n})$.\n Similarly to the proof of \\eqref{eq:38}, we obtain\n $|z_{n}(t)|_{\\dot{H}^{2}} =\n |z_{0,y}^{n,\\psi}(t)|_{\\dot{H}^{2}}\n \\leq C(1+|y|_{\\dot{H}^{2}}),t \\in [0,T]$, and thus\n \\begin{align*}\n &\\big|Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n -F_{n}(z_{n}(t))\\big)\\big|^{2}\n \\leq\n C\\big|(-A_{n})\\big|_{\\mathcal{L}(H_{n})}^{2}\n \\big(1+|y|_{\\dot{H}^{2}}^{2}\\big)^{2}|y|_{\\dot{H}^{2}}^{2}\n <\n +\\infty.\n \\end{align*}\n By $\\lim\\limits_{\\tau \\to 0}\n \\big|z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n -z_{n}(t) \\big| = 0, t \\in [0,T]$, we have\n \\begin{align*}\n \\lim\\limits_{\\tau \\to 0}\n \\big|Q_{n}^{-\\frac12}\n \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n -F_{n}(z_{n}(t))\\big)\\big|^{2}\n \\leq\n C\\big|(-A_{n})\\big|_{\\mathcal{L}(H_{n})}^{2}\n \\lim\\limits_{\\tau \\to 0}\n \\big|z_{n}(\\tau \\lfloor t\/\\tau \\rfloor) - z_{n}(t)\\big|^{2}\n =\n 0.\n \\end{align*}\n It follows from the bounded convergence theorem that\n \\begin{equation}\\label{eq:410410410\n \\lim\\limits_{\\tau \\to 0}|I_{0,T}^{n,y}(z_{n})\n -I_{0,T}^{n,\\tau,y}(z_{n})| = 0,\n \\end{equation}\n and consequently that there exists $\\tau > 0$ such that\n \\begin{equation}\\label{eq:part222\n \\big|I_{0,T}^{n,y}(z_{n})\n -I_{0,T}^{n,\\tau,y}(z_{n})\\big|\n \\leq\n \\frac{\\kappa}{2}.\n \\end{equation}\n For $n,\\tau$ given by \\eqref{eq:part111} and \\eqref{eq:part222}, we set $z_{n,\\tau} := z_{n}$ and use the triangle inequality to complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsection{LDP of invariant measures of spatio-temporal discretization}\n\\label{subsec:fullinvameaLDP\n\n\n\nIn this subsection, we study the LDP of invariant measures of the spatio-temporal discretization. The following lemma gives the uniform boundedness of numerical solutions, which ensures the existence of the invariant measures of the full discretization. We refer the interested readers to \\cite{chen2017approximation,hong2017numerical,\nhong2019invariant,cui2021weak} for the investigation on the invariant measures of the full discretzations for SPDEs.\n\n\n\n\n\n\\begin{Lemma}\\label{lem:uniformbound2ordermoment}\n Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. If $\\tau \\leq \\tau_{0}$ with $\\tau_{0}$ satisfying\n $\\frac{e^{\\lambda_{1}\\tau_{0}}-1}{\\lambda_{1}\\tau_{0}} = \\frac{\\lambda_{1}+L_{F}}{2L_{F}}$, then there exists $C > 0$ independent of $t$ such that\n \\begin{equation}\\label{eq:Y0vartHn\n \\sup\\limits_{t \\geq 0}\n \\mathbb{E}\\big[|Y_{y}^{\\varepsilon,n}(t)|^{2}\\big]\n \\leq\n C.\n \\end{equation}\n\\end{Lemma}\n\n\n\n\\begin{proof}\n We rewrite \\eqref{eq:EE} as\n \\begin{align*}\n Y_{y,m}^{\\varepsilon,n}\n =&\n E_{n}^{m}(\\tau)Y_{y,0}^{\\varepsilon,n}\n +\n \\int_{0}^{t_{m}}E_{n}(t_{m}-s)\n F_{n}(Y_{y,\\lfloor s\/\\tau \\rfloor}^{\\varepsilon,n})\n \\diff{s}\n +\n \\varepsilon \\int_{0}^{t_{m}} E_{n}(t_{m}-s) Q_{n}^{\\frac12}\\diff{W_{n}(s)}.\n \\end{align*}\n Noting $\\{\\Gamma^{n}(t)\\}_{t \\geq 0}$ given by \\eqref{eq:Gammant} and setting $\\bar{Y}_{y,m}^{\\varepsilon,n} := Y_{y,m}^{\\varepsilon,n} - \\varepsilon\\Gamma^{n}(t_{m})$, it follows that\n \\begin{align*}\n \\bar{Y}_{y,m}^{\\varepsilon,n}\n =&\n E_{n}^{m}(\\tau)\\bar{Y}_{y,0}^{\\varepsilon,n}\n +\n \\int_{0}^{t_{m}}E_{n}(t_{m}-s)\n F_{n}(\\bar{Y}_{y,\\lfloor s\/\\tau \\rfloor}^{\\varepsilon,n}\n +\\varepsilon\\Gamma^{n}(\\tau \\lfloor s\/\\tau \\rfloor))\n \\diff{s},\n \\quad\n \\bar{Y}_{y,0}^{\\varepsilon,n} = y\n \\end{align*}\n and thus\n \\begin{align*}\n \\bar{Y}_{y,m}^{\\varepsilon,n}\n =&\n E_{n}(\\tau)\\bar{Y}_{y,m-1}^{\\varepsilon,n}\n +\n (-A_{n})^{-1}(I-E_{n}(\\tau))\n F_{n}(\\bar{Y}_{y,m-1}^{\\varepsilon,n}\n +\\varepsilon\\Gamma^{n}(t_{m-1})).\n \\end{align*}\n Applying $|E_{n}(\\tau)|_{\\mathcal{L}(H_{n})} \\leq e^{-\\lambda_{1}\\tau}$ and\n $|(-A_{n})^{-1}(I-E_{n}(\\tau))|_{\\mathcal{L}(H_{n})} \\leq (1-e^{-\\lambda_{1}\\tau})\/\\lambda_{1} \\leq \\tau$ yields\n \\begin{align*}\n |\\bar{Y}_{y,m}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n \\leq&\n e^{-\\lambda_{1}\\tau}\n |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n +\n \\tau \\frac{1-e^{-\\lambda_{1}\\tau}}{\\lambda_{1}\\tau}\n |F_{n}(\\bar{Y}_{y,m-1}^{\\varepsilon,n}\n +\\varepsilon\\Gamma^{n}(t_{m-1}))|_{L^{2}(\\Omega;H)}\n \\\\\\leq&\n \\Big(e^{-\\lambda_{1}\\tau}\n +\n \\tau L_{F}\\frac{1-e^{-\\lambda_{1}\\tau}}{\\lambda_{1}\\tau}\\Big)\n |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n +\n \\tau L_{F}\\frac{1-e^{-\\lambda_{1}\\tau}}{\\lambda_{1}\\tau}\n |\\varepsilon\\Gamma^{n}(t_{m-1})|_{L^{2}(\\Omega;H)}.\n \\end{align*}\n Recalling $\\sup\\limits_{t \\geq 0}\n |\\Gamma^{n}(t)|_{L^{2}(\\Omega;H)} \\leq C$ in \\eqref{eq:GammanC0tHn2} and\n $\\tau \\leq \\tau_{0}$ with $\\tau_{0}$ satisfying\n $\\frac{e^{\\lambda_{1}\\tau_{0}}-1}{\\lambda_{1}\\tau_{0}} = \\frac{\\lambda_{1}+L_{F}}{2L_{F}}$,\n we get\n \\begin{align*}\n |\\bar{Y}_{y,m}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n \\leq&\n \\Big(e^{-\\lambda_{1}\\tau}\n +\n \\tau L_{F}e^{-\\lambda_{1}\\tau}\n +\n \\tau\\frac{\\lambda_{1}-L_{F}}{2}e^{-\\lambda_{1}\\tau}\\Big)\n |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n +\n C\\tau\n \\\\=&\n \\Big(1 + \\tau\\frac{\\lambda_{1}+L_{F}}{2}\\Big)\n e^{-\\lambda_{1}\\tau}\n |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n +\n C\\tau\n \\\\\\leq&\n e^{-\\frac{\\lambda_{1}-L_{F}}{2}\\tau}\n |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n +\n C\\tau\n \\\\\\leq&\n e^{-\\frac{\\lambda_{1}-L_{F}}{2}m\\tau}|y|\n +\n C\\tau\\frac{e^{\\frac{\\lambda_{1}-L_{F}}{2}\\tau}}\n {e^{\\frac{\\lambda_{1}-L_{F}}{2}\\tau}-1}\n \\\\\\leq&\n |y|\n +\n 2C\\frac{e^{\\frac{\\lambda_{1}-L_{F}}{2}\\tau_{0}}}\n {\\lambda_{1}-L_{F}}.\n \\end{align*}\n Thus we assert that there exists $C > 0$ independent of $m \\in \\mathbb{N}$ such that $\\sup\\limits_{m \\in \\mathbb{N}}|Y_{y,m}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)} \\leq C$.\n \n It follows from \\eqref{eq:4242} that\n \\begin{align*}\n |Y_{y}^{\\varepsilon,n}(t)|_{L^{2}(\\Omega;H)}\n \\leq&\n |Y_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n +\n |F_{n}(Y_{y,m-1}^{\\varepsilon,n})|_{L^{2}(\\Omega;H)}\n +\n |\\varepsilon \\Gamma^{n}(t)|_{L^{2}(\\Omega;H)}\n \\leq\n C\n \\end{align*}\n with $C$ being independent of $t$. Thus we complete the proof.\n\\end{proof}\n\n\n\n\n\nBy Lemma \\ref{lem:uniformbound2ordermoment} and \\cite[Proposition 7.10]{da2006introduction}, the family of probability measures $\\{\\mu_{t}^{\\varepsilon,n,\\tau}\\}_{t > 0}$, defined by\n $$\n \\mu_{t}^{\\varepsilon,n,\\tau}(B)\n :=\n \\frac{1}{t} \\int_{0}^{t}\n \\P\\big(Y_{0}^{\\varepsilon,n}(s) \\in B\\big)\n \\diff{s},\n \\quad B \\in \\mathcal{B}(H_{n}), t > 0,\n $$\nis tight. It follows from the Krylov--Bogoliubov theorem (\\cite[Theorem 7.1]{da2006introduction}) that there exists $\\{t_{i}\\}_{i \\in \\mathbb{N}^{+}} \\uparrow +\\infty$ (possibly depending on $\\varepsilon$) such that the sequence $\\{\\mu_{t_{i}}^{\\varepsilon,n,\\tau}\\}_{i \\in \\mathbb{N}^{+}}$ converges weakly to some probability measure $\\mu^{\\varepsilon,n,\\tau}$ on $(H_{n},\\mathcal{B}(H_{n}))$, which is invariant for \\eqref{eq:saptiotemporalconteq}. Following the procedures\nin Subsection \\ref{subsec:semiinvameaLDPs}, we establish the LDP for $\\{\\mu^{\\varepsilon,n,\\tau}\\}_{\\varepsilon > 0}$ on $H_{n}$ with the rate function $V^{n,\\tau}$ defined by\n\\begin{equation}\\label{eq:Vntauy\n V^{n,\\tau}(u)\n =\n \\inf\\{I_{0,T}^{n,\\tau,0}(z) : T > 0, z \\in C([0,T];H_{n}), z(0) = 0, z(T) = u \\},\n \\quad u \\in H_{n}.\n\\end{equation}\n\n\n\\begin{Theorem}\n \n Under assumptions in Lemma \\ref{lem:uniformbound2ordermoment}, $\\{\\mu^{\\varepsilon,n,\\tau}\\}_{\\varepsilon > 0}$ satisfies an LDP on $H_{n}$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $V^{n,\\tau}$ given by \\eqref{eq:Vntauy}.\n \n\\end{Theorem}\n\n\nWe give the definition of weakly asymptotical preservation for the LDP of invariant measures by a numerical method to characterize the error between $V^{n,\\tau}$ and $V$.\n\n\n\n\n\n\\begin{Definition}\n We say that\n the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $u \\in \\mathcal{D}_{V}$, there exist $n \\in \\mathbb{N}^{+}$, $\\tau > 0$ and $u_{n,\\tau} \\in H_{n}$ such that\n $$\n |u-u_{n,\\tau}| < \\kappa,\n \\qquad\n |V(u)-V^{n,\\tau}(u_{n,\\tau})| < \\kappa.\n $$\n\\end{Definition}\n\n\n\n\n\n\\begin{Theorem}\\label{fullinvameaLDPasypre\n Suppose that assumptions in Lemma \\ref{lem:uniformbound2ordermoment} hold.\n \n If\n $F \\equiv \\textbf{0}$,\n then\n the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\n\\begin{proof}\n For any $\\kappa > 0$ and $u \\in \\mathcal{D}_{V}$, Theorem \\ref{th:semiinvameaLDPasympre} shows that there exist\n $n \\in \\mathbb{N}^{+}$ and $u_{n} \\in H_{n}$ such that\n \\begin{equation}\\label{eq:invapart111\n |u-u_{n}| < \\frac{\\kappa}{2} < \\kappa,\n \\qquad\n |V(u)-V^{n}(u_{n})| < \\frac{\\kappa}{2}.\n \\end{equation}\n Then the definition of $V^{n}(u_{n})$ in \\eqref{eq:Vny} ensures that for any $\\kappa > 0$ there exist $T_{n,\\kappa} > 0, z_{n,\\kappa} \\in C([0,T_{n,\\kappa}];H_{n}), z_{n,\\kappa}(0) = 0$ and $z_{n,\\kappa}(T_{n,\\kappa}) = u_{n}$ such that\n \\begin{equation}\\label{eq:518\n I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n \\leq\n V^{n}(u_{n}) + \\frac{\\kappa}{4}\n <\n +\\infty,\n \\end{equation}\n which implies $z_{n,\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{n,\\kappa})$ by \\eqref{eq:I0Tnyz}. According to \\eqref{eq:Vntauy}, we have\n \\begin{equation}\\label{eq:519\n V^{n,\\tau}(u_{n})\n \\leq\n I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa}).\n \\end{equation}\n As \\eqref{eq:410410410} in Theorem \\ref{th:fullLDPasympre} ensures\n $\n \\lim\\limits_{\\tau \\to 0}I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa})\n =\n I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})$, i.e.,\n there exists $\\tau > 0$ such that\n \\begin{equation}\\label{eq:520\n \\big|I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa})\n -\n I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\\big|\n \\leq\n \\frac{\\kappa}{4},\n \\end{equation}\n it follows from \\eqref{eq:518}, \\eqref{eq:519} and \\eqref{eq:520} that\n \\begin{equation}\\label{eq:521\n V^{n,\\tau}(u_{n})\n \\leq\n I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa})\n \\leq\n I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n +\n \\frac{\\kappa}{4}\n \\leq\n V^{n}(u_{n}) + \\frac{\\kappa}{2}.\n \\end{equation}\n It remains to show $V^{n}(u_{n}) \\leq V^{n,\\tau}(u_{n}) + \\frac{\\kappa}{2}$ for the above given $\\tau > 0$. The definition of $V^{n,\\tau}(u_{n})$ in \\eqref{eq:Vntauy} implies that for $\\kappa > 0$ there exists $T_{n,\\tau,\\kappa} > 0, z_{n,\\tau,\\kappa} \\in C([0,T_{n,\\tau,\\kappa}];H_{n}), z_{n,\\tau,\\kappa}(0) = 0$ and $z_{n,\\tau,\\kappa}(T_{n,\\tau,\\kappa}) = u_{n}$ such that\n \\begin{equation}\\label{eq:522\n I_{0,T_{n,\\tau,\\kappa}}^{n,\\tau,0}(z_{n,\\tau,\\kappa})\n \\leq\n V^{n,\\tau}(u_{n}) + \\frac{\\kappa}{2}\n \\leq\n V^{n}(u_{n}) + \\kappa\n <\n +\\infty,\n \\end{equation}\n which together with \\eqref{eq:I0TnMyz} implies $z_{n,\\tau,\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{n,\\tau,\\kappa})$.\n The definition of $V^{n}(u_{n})$ in \\eqref{eq:Vny} yields\n \\begin{equation}\\label{eq:523\n V^{n}(u_{n})\n \\leq\n I_{0,T_{n,\\tau,\\kappa}}^{n,0}(z_{n,\\tau,\\kappa}).\n \\end{equation}\n Taking advantaging of $F \\equiv \\textbf{0}$, we have\n \\begin{equation}\\label{eq:524\n I_{0,T_{n,\\tau,\\kappa}}^{n,0}(z_{n,\\tau,\\kappa})\n =\n \\frac{1}{2}\n \\int_{0}^{T_{n,\\tau,\\kappa}}\n \\Big|Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z_{n,\\tau,\\kappa}(t)}}{\\diff{t}}\n -\n A_{n}z_{n,\\tau,\\kappa}(t)\n \n \\Big)\n \\Big|^{2}\n \\diff{t}\n =\n I_{0,T_{n,\\tau,\\kappa}}^{n,\\tau,0}(z_{n,\\tau,\\kappa}).\n \\end{equation}\n This together with \\eqref{eq:522}, \\eqref{eq:523} and \\eqref{eq:524} leads to\n \\begin{equation}\\label{eq:525\n V^{n}(u_{n})\n \\leq\n I_{0,T_{n,\\tau,\\kappa}}^{n,0}(z_{n,\\tau,\\kappa})\n =\n I_{0,T_{n,\\tau,\\kappa}}^{n,\\tau,0}(z_{n,\\tau,\\kappa})\n \\leq\n V^{n,\\tau}(u_{n}) + \\frac{\\kappa}{2}.\n \\end{equation}\n Then \\eqref{eq:521} and \\eqref{eq:525} show\n \\begin{equation}\\label{eq:invapart222\n |V^{n}(u_{n})-V^{n,\\tau}(u_{n})| \\leq \\frac{\\kappa}{2}.\n \\end{equation}\n For $n,\\tau$ given by \\eqref{eq:invapart111} and \\eqref{eq:520}, we set $u_{n,\\tau} := u_{n}$. By \\eqref{eq:invapart111}, \\eqref{eq:invapart222} and the triangle inequality, we complete the proof.\n\\end{proof}\n\n\n\\begin{Remark}\n When $F \\colon H \\to H$ is a linear mapping, i.e., $F(u) = Bu$ for some $B \\in \\mathcal{L}(H)$ with $|B|_{\\mathcal{L}(H)} < \\lambda_{1}$, we can redefine the unbounded linear operator $A$ by $A+B$ to vanish $F$,\n as well as the numerical discretizations. Repeating procedure in Theorems \\ref{th:semiinvameaLDPasympre} and \\ref{fullinvameaLDPasypre} still leads to the weakly asymptotical preservation for the corresponding LDPs by the numerical approximations.\n\\end{Remark}\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}