Itai Benjamini, Noam Berger, Ariel Yadin
We provide an estimate, sharp up to poly-logarithmic factors, of the asymptotically almost sure mixing time of the graph created by long-range percolation on the cycle of length N (Z/NZ). While it is known that the almost sure diameter drops from linear to poly-logarithmic as the exponent s decreases below 2, the almost sure mixing time drops from N^2 only to N^(s-1) (up to poly-logarithmic factors).
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We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in $\Z^d$. We show that for $d \geq 2$ and $p > p_c(\Z^d)$, the mixing time of simple random walk on the largest cluster inside $\{-n,...,n\}^d$ is $Θ(n^2)$ - thus the mixing time is robust up to constant factor.
Mixing rate exponent of planar Fortuin-Kasteleyn percolation
该研究定义了共形环系综（CLE）对应的混合速率指数，在FK渗流收敛于CLE的假设下证明二者指数一致，并借助刘维尔量子引力与CLE的耦合推导出CLE$_κ$的混合速率指数公式，解答了前人提出的相关问题。
Mixing times of spin systems on dynamical percolation
研究d维环面上动态渗流中最近邻Glauber动力学的混合时间，证明当p<p_c(d)且λ足够小时，混合时间阶为log N/λ。
We introduce a new framework for analyzing Glauber dynamics for the Ising model. The traditional approach for obtaining sharp mixing results has been to appeal to estimates on spatial properties of the stationary measure from within a multi-scale analysis of the dynamics. Here we propose to study these simultaneously by examining "information percolation" clusters in the space-time slab. Using this framework, we obtain new results for the Ising model on $(\mathbb{Z}/n\mathbb{Z})^d$ throughout the high temperature regime: total-variation mixing exhibits cutoff with an $O(1)$-window around the t…
Bounds for the diameter and for the expansion of long-range percolation clusters on the cycle $\Z / N\Z$ are given.
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate μ while at the same time a random walker moves on G at rate 1 but only along edges which are open. On the d-dimensional torus with side length n, we prove that in the subcritical regime, the mixing times for both the full system and the random walker are n^2/μ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice Z^d…
Assign to each vertex of the one-dimensional torus i.i.d. weights with a heavy-tail of index $τ-1>0$. Connect then each couple of vertices with probability roughly proportional to the product of their weights and that decays polynomially with exponent $α>0$ in their distance. The resulting graph is called scale-free percolation. The goal of this work is to study the mixing time of the simple random walk on this structure. We depict a rich phase diagram in $α$ and $τ$. In particular we prove that the presence of hubs can speed up the mixing of the chain. We use different techniques for each pha…
For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further…
We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $μ=μ_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(\mathbb{Z}^d)$, the (annealed) mixing time of the walk is $Θ(n^2/μ)$, and it was conjectured that in the supercritical case $p>p_c(\mathbb{Z}^d)$, the mixing time is $Θ(n^2+1/μ)$; here the implie…
Mixing time and isoperimetry of long-range percolation in the diffusive regime
该研究针对扩散 regime 下的长程渗流，证明超临界长程渗流巨连通分支上随机游走的混合时间量级为 n²，核心是通过巨连通分支内大连通集的 d 维等周不等式建立混合时间上界。
Strict monotonicity of critical points in independent long-range percolation models
该研究针对局部有限顶点传递图上的独立长程渗流模型，通过耦合思想证明临界点关于连接函数局部扰动的严格单调性，在更少假设下改进了经典方法的结果，且适用于无向和有向边渗流模型。
Mixing times for the TASEP on the circle
本文研究了环上TASEP的混合时间，证明其与N²和min(k,N−k)相关，并揭示了该过程不存在临界现象。
Information percolation is a new method for analyzing stochastic spin systems through classifying and controlling the clusters of information-flow in the space-time slab. It yielded sharp mixing estimates (cutoff with an $O(1)$-window) for the Ising model on $Z^d$ up to the critical temperature, as well as results on the effect of initial conditions on mixing. In this expository note we demonstrate the method on lattices (more generally, on any locally-finite transitive graph) at very high temperatures.
This is a (long) survey about applications of percolation theory in equilibrium statistical mechanics. The chapters are as follows: 1. Introduction 2. Equilibrium phases 3. Some models 4. Coupling and stochastic domination 5. Percolation 6. Random-cluster representations 7. Uniqueness and exponential mixing from non-percolation 8. Phase transition and percolation 9. Random interactions 10. Continuum models
Network-based analysis of stochastic SIR epidemic models with random and proportionate mixing
本文概述流行病渗流网络理论，并将其应用于分析无向接触网络及随机与比例混合下的随机SIR流行病模型，证明该理论能准确预测暴发规模与概率，并为封闭群体中的随机SIR模型分析提供通用框架。
Consider independent long range percolation on $\mathbf{Z}^2$, where horizontal and vertical edges of length $n$ are open with probability $p_n$. We show that if $\limsup_{n\to\infty}p_n>0,$ then there exists an integer $N$ such that $P_N(0\leftrightarrow \infty)>0$, where $P_N$ is the truncated measure obtained by taking $p_{N,n}=p_n$ for $n \leq N$ and $p_{N,n}=0$ for all $n> N$.
In this short note we consider mixed short-long range independent bond percolation models on $\Z^{k+d}$. Let $p_{uv}$ be the probability that the edge $(u,v)$ will be open. Allowing a $x,y$-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity $τ_{xy}$ is governed by the probability $p_{xy}$. The result holds up to the critical point.
We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-α}$ with $α>0$. For random walk in any dimension $d$ and for self-avoiding walk and critical/subcritical oriented percolation above the common upper-critical dimension $d_{\mathrm{c}}\equiv2(α\wedge2)$, we prove large-$t$ asymptotics of the gyration radius, which is the average end-to-end distance of random walk/self-avoiding walk of length $t$ or the average spatial size of an oriente…
We consider the ensemble of $N\times N$ real random symmetric matrices $H_N^{(R)}$ obtained from the determinant form of the Ihara zeta function associated to random graphs $Γ_N^{(R)}$ of the long-range percolation radius model with the edge probability determined by a function $ϕ(t)$. We show that the normalized eigenvalue counting function of $H_N^{( R)}$ weakly converges in average as $N,R\to\infty$, $R=o(N)$ to a unique measure that depends on the limiting average vertex degree of $Γ_N^{(R)}$ given by $ϕ_1 = \int ϕ(t) dt$. This measure converges in the limit of infinite $ϕ_1$ to a shift of…
We prove that the heat kernel on the infinite Bernoulli percolation cluster in Z^d almost surely decays faster than t^{-d/2}. We also derive estimates on the mixing time for the random walk confined to a finite box. Our approach is based on local isoperimetric inequalities.
We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)
Consider a long-range percolation model on $\mathbb{Z}^d$ where the probability that an edge $\{x,y\} \in \mathbb{Z}^d \times \mathbb{Z}^d$ is open is proportional to $\|x-y\|_2^{-d-α}$ for some $α>0$ and where $d > 3 \min\{2,α\}$. We prove that in this case the one-arm exponent equals $ \min\{4,α\}/2$. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter $α$ when $α=4$.
Continuity of the critical value and a shape theorem for long-range percolation
研究了长程渗透中临界值的连续性及无限簇中可达点的形状定理，揭示了模型精确规格对临界值的影响。
We study limit laws for simple random walks on supercritical long range percolation clusters on $\Z^d, d \geq 1$. For the long range percolation model, the probability that two vertices $x, y$ are connected behaves asymptotically as $\|x-y\|_2^{-s}$. When $s\in(d, d+1)$, we prove that the scaling limit of simple random walk on the infinite component converges to an $α$-stable Lévy process with $α= s-d$ establishing a conjecture of Berger and Biskup. The convergence holds in both the quenched and annealed senses. In the case where $d=1$ and $s>2$ we show that the simple random walk converges to…
In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any $d \geq 1$ and for any exponent $s \in (d, (d+2) \wedge 2d)$ giving the rate of decay of the percolation process, we show that the return probability decays like $t^{-\ffrac{d}{s-d}}$ up to logarithmic corrections, where $t$ denotes the time the walk is run. Moreover, our methods also yield generalized bounds on the spectral gap of the dynamics and on the diameter of the largest component in a box. Besides its intrinsic interest,…
We consider random walk on dynamical percolation on the discrete torus $\mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(\mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index α>0 converges to e^{-C|k|^{α\wedge2}} for some C\in(0,\infty) above the upper-critical dimension 2(α\wedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at α=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficien…
We consider oriented percolation on Z^d times Z_+ whose bond-occupation probability is pD(...), where p is the percolation parameter and D is a probability distribution on Z^d. Suppose that D(x) decays as |x|^{-d-α} for some α>0. We prove that the two-point function obeys an infrared bound which implies that various critical exponents take on their respective mean-field values above the upper-critical dimension 2\min{α,2}. We also show that, for every k, the Fourier transform of the normalized two-point function at time n, with a proper spatial scaling, has a convergent subsequence to exp(-c|k…
We consider the following long range percolation model: an undirected graph with the node set $\{0,1,...,N\}^d$, has edges $(\x,\y)$ selected with probability $\approx β/||\x-\y||^s$ if $||\x-\y||>1$, and with probability 1 if $||\x-\y||=1$, for some parameters $β,s>0$. This model was introduced by Benjamini and Berger, who obtained bounds on the diameter of this graph for the one-dimensional case $d=1$ and for various values of $s$, but left cases $s=1,2$ open. We show that, with high probability, the diameter of this graph is $Θ(\log N/\log\log N)$ when $s=d$, and, for some constants $0<η_1<…
Consider an independent site percolation model on $\Z^d$, with parameter $p \in (0,1)$, where all long range connections in the axes directions are allowed. In this work we show that given any parameter $p$, there exists and integer $K(p)$ such that all binary sequences (words) $ξ\in \{0,1\}^{\N}$ can be seen simultaneously, almost surely, even if all connections whose length is bigger than $K(p)$ are suppressed. We also show some results concerning the question how $K(p)$ should scale with $p$ when $p$ goes to zero. Related results are also obtained for the question of whether or not almost a…
This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding "long edges (short cuts)" randomly to the largest percolation cluster, we obtain a random graph $\mathscr G_n$. In the present paper, we first prove that the diameter of $\mathscr G_n$ grows at most polynomially fast in $\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\mathscr G_n$ possesses the rapid mixing property, namely, the …
We prove up-to-constants bounds on the two-point function (i.e., point-to-point connection probabilities) for critical long-range percolation on the $d$-dimensional hierarchical lattice. More precisely, we prove that if we connect each pair of points $x$ and $y$ by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $0<α<d$ is fixed and $β\geq 0$ is a parameter, then the critical two-point function satisfies \[ \mathbb{P}_{β_c}(x\leftrightarrow y) \asymp \|x-y\|^{-d+α} \] for every pair of distinct points $x$ and $y$. We deduce in particular that the model has mean-field critical behavi…
Consider an independent site percolation model with parameter $p \in (0,1)$ on $\Z^d,\ d\geq 2$ where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to each coordinate axis. We show that the percolation threshold of such model converges to $p_c(\Z^{2d})$ when $k$ goes to infinity, the percolation threshold for ordinary (nearest neighbour) percolation on $\Z^{2d}$. We also generalize this result for models whose long range bonds have several lengths.
Consider long-range Bernoulli percolation on $\mathbb{Z}^d$ in which we connect each pair of distinct points $x$ and $y$ by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$, where $α>0$ is fixed and $β\geq 0$ is a parameter. We prove that if $0<α<d$ then the critical two-point function satisfies \[ \frac{1}{|Λ_r|}\sum_{x\in Λ_r} \mathbf{P}_{β_c}(0\leftrightarrow x) \preceq r^{-d+α} \] for every $r\geq 1$, where $Λ_r=[-r,r]^d \cap \mathbb{Z}^d$. In other words, the critical two-point function on $\mathbb{Z}^d$ is always bounded above on average by the critical two-point function on the hiera…
We consider instances of long-range percolation on $\mathbb Z^d$ and $\mathbb R^d$, where points at distance $r$ get connected by an edge with probability proportional to $r^{-s}$, for $s\in (d,2d)$, and study the asymptotic of the graph-theoretical (a.k.a. chemical) distance $D(x,y)$ between $x$ and $y$ in the limit as $|x-y|\to\infty$. For the model on $\mathbb Z^d$ we show that, in probability as $|x|\to\infty$, the distance $D(0,x)$ is squeezed between two positive multiples of $(\log r)^Δ$, where $Δ:=1/\log_2(1/γ)$ for $γ:=s/(2d)$. For the model on $\mathbb R^d$ we show that $D(0,xr)$ is,…
We study a general class of percolation models in Euclidean space including long-range percolation, scale-free percolation, the weight-dependent random connection model and several other previously investigated models. Our focus is on the weak decay regime, in which inter-cluster long-range connection probabilities fall off polynomially with small exponent, and for which we establish several structural properties. Chief among them are the continuity of the bond percolation function and the transience of infinite clusters.
For a general class of percolation models with long-range correlations on $\mathbb Z^d$, $d\geq 2$, introduced in arXiv:1212.2885, we establish regularity conditions of Barlow arXiv:math/0302004 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincaré inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of arX…
We study mixing times for the totally asymmetric simple exclusion process (TASEP) on a segment of size $N$ with open boundaries. We focus on the maximal current phase, and prove that the mixing time is of order $N^{3/2}$, up to logarithmic corrections. In the triple point, where the TASEP with open boundaries approaches the Uniform distribution on the state space, we show that the mixing time is precisely of order $N^{3/2}$. This is conjectured to be the correct order of the mixing time for a wide range of particle systems with maximal current. Our arguments rely on a connection to last-passag…
We show quenched large deviations for the simple random walk on a certain class of percolations with long-range correlations. This class contains the supercritical Bernoulli percolations, the model considered by Drewitz, R'ath and Sapozhnikov and the random-cluster model up to the slab critical point. Our result is an extension of Kubota's result for the supercritical Bernoulli percolations. We also state a shape theorem for the chemical distance, which is an extension of Garet and Marchand's result for the supercritical Bernoulli percolations.
Consider an independent site percolation model on $\Z^d,\ d\geq 2$, with parameter $p \in (0,1)$, where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to some coordinate axis. We show that the percolation probability is a non-decreasing function of $k$. We also generalize this result for models whose long range bonds have several lengths.
We consider different problems within the general theme of long-range percolation on oriented graphs. Our aim is to settle the so-called truncation question, described as follows. We are given probabilities that certain long-range oriented bonds are open; assuming that the sum of these probabilities is infinite, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. We give some conditions in which the answer is affirmative. We also translate some of our results on oriented percolation to the cont…
We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-β\|x-y\|^{-d-α})$. It is a theorem of Noam Berger (CMP, 2002) that if $0<α<d$ then there is no infinite cluster at the critical parameter $β_c$. We give a new, quantitative proof of this theorem establishing the power-law upper bound \[ \mathbf{P}_{β_c}\bigl(|K|\geq n\bigr) \leq C n^{-(d-α)/(2d+α)} \] for every $n\geq 1$, where $K$ is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli perco…
The study of random graphs has become very popular for real-life network modeling such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the lattice $\mathbb Z^d$, $d\ge1$, is a particular attractive example of a random graph model because it fulfills several stylized facts of real-life networks. For this model various geometric properties such as the percolation behavior, the degree distribution and graph distances have been analyzed. In the present paper we complement the picture about graph distances. Moreover, we prove continuity …
First passage percolation with long-range correlations and applications to random Schrödinger operators
本文研究长程相关环境中的首次通过渗流，证明时间常数为正、距离可比及形状定理，并将结果用于随机电导模型的热核上界与 Green 函数衰减估计。
We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$, we show that the graph distance is at least linear with the Euclidean distance.
Recently, by introducing the notion of cumulatively merged partition, Ménard and Singh provide a sufficient condition on graphs ensuring that the critical value of the contact process is positive. In this note, we show that the one-dimensional long range percolation with high exponent satisfies their condition and thus the contact process exhibits a non-trivial phase transition.
Dimension dependence of critical phenomena in long-range percolation
本文面向数学读者概述了长程渗流等统计力学模型临界行为的维度依赖性相关猜想，包括上临界维度、长短程区域转变等内容，并调研了该领域近期重要进展。
We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that this probabilities are not summable, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. This question is still open if the set of vertices is $\Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hilário, de Lima, Valesin, Journ. Stat. Phys. {\…
We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-β^{-k} α)$, independently of all other edges. For fixed $β$, we show that the critical value $α_c(β)$ is non-trivial if and only if $N < β< N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $α_c(β)$ as a function of $β$. This means that the phase diagram of this model is well understood.
We study the normalized eigenvalue counting measure dσof matrices of long-range percolation model. These are (2n+1)\times (2n+1) random real symmetric matrices H=\{H(i,j)\}_{i,j} whose elements are independent random variables taking zero value with probability 1-ψ[(i-j)/b], b\in \mathbb{R}^{+}, where ψis an even positive function ψ(t)\le{1} vanishing at infinity. It is shown that if the third moment of \sqrt{b}H(i,j), i\leq{j} is uniformly bounded then the measure dσ:=dσ_{n,b} weakly converges in probability in the limit n,b\to\infty, b=o(n) to the semicircle (or Wigner) distribution. The pro…
We study the spectral properties of matrices of long-range percolation model. These are N\times N random real symmetric matrices H=\{H(i,j)\}_{i,j} whose elements are independent random variables taking zero value with probability 1-ψ((i-j)/b), b\in \mathbb{R}^{+}, where $ψ$ is an even positive function with ψ(t)\le{1} and vanishing at infinity. We study the resolvent G(z)=(H-z)^{-1}, Imz\neq{0} in the limit N,b\to\infty, b=O(N^α), 1/3<α<1 and obtain the explicit expression T(z_{1},z_{2}) for the leading term of the correlation function of the normalized trace of resolvent g_{N,b}(z)=N^{-1}Tr …
We consider oriented long-range percolation on a graph with vertex set $\mathbb{Z}^d \times \mathbb{Z}_+$ and directed edges of the form $\langle (x,t), (x+y,t+1)\rangle$, for $x,y$ in $\mathbb{Z}^d$ and $t \in \mathbb{Z}_+$. Any edge of this form is open with probability $p_y$, independently for all edges. Under the assumption that the values $p_y$ do not vanish at infinity, we show that there is percolation even if all edges of length more than $k$ are deleted, for $k$ large enough. We also state the analogous result for a long-range contact process on $\mathbb{Z}^d$.
Uniqueness and dimension for the geodesic of the critical long-range percolation metric
本文在临界长程渗流随机度量已构建的基础上，证明了两固定点间测地线的唯一性（关键要素为度量分布的连续性），并确定了测地线的Hausdorff维数。
Long-range contact process and percolation on a random lattice
本文研究随机格上长程有向渗流与接触过程的相变，通过引入由分布N决定的顶点作用范围，证明了两类模型相变存在性依赖于分布N的条件。
Scaling limits for random walks on long range percolation clusters
研究整数格上超临界长程渗流簇的简单随机游动，证明其标度极限收敛于各向同性 alpha-稳定 Levy 过程，且收敛在 quenched 和 annealed 意义下均成立。
We study the long range percolation model on $\mathbb{Z}$ where sites $i$ and $j$ are connected with probability $β|i-j|^{-s}$. Graph distances are now well understood for all exponents $s$ except in the case $s=2$ where the model exhibits non-trivial self-similar scaling. Establishing a conjecture of Benjamini and Berger \cite{BenBer:01}, we prove that the typical distance from site 0 to $n$ grows as a power law $n^{θ(β)}$ up to a multiplicative constant for some exponent $0<θ(β)<1$ as does the diameter of the graph on a box of length $n$.
Uniqueness of the critical long-range percolation metrics
本文研究 $\mathbb{Z}^d$ 上临界长程渗流的随机度量，证明其已有子序列标度极限可由一组自然公理唯一刻画。
The critical curve of long-range percolation on oriented trees
该研究在有向树上分析长程渗透模型，通过特征多项式研究得到临界曲线的性质及新界，揭示其连续性和凸性特征。
Graph distances of continuum long-range percolation
本文研究了基于 Poisson 点过程的连续长程渗流模型中的图论距离，证明了该随机图模型在 $s=d$ 和 $s=2d$ 处发生相变，与经典长程渗流类似。
From local giants to locality in long-range percolation
研究在特定多项式增长传递图上长程渗流的Schramm局部性猜想类似物，通过构建重整化方案证明相关结果，包括渗流概率联合连续性、超临界锐度等，还得出大数定律，回答了特定问题。
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