Massimo Campanino, Michele Gianfelice
We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on $\mathbb{Z}^{d},\,d\geq2$ in the limit where their distances tend to infinity.
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We prove that for Bernoulli percolation on a graph $\mathbb{Z}^2\times\{0,\dots,k\}$ ($k\ge 0$), there is no infinite cluster at criticality, almost surely. The proof extends to finite range Bernoulli percolation models on $\mathbb{Z}^2$ which are invariant under $π/2$-rotation and reflection.
Two vertices are said to be finitely connected if they belong to the same cluster and this cluster is finite. We derive sharp asymptotics for finite connection probabilities for supercritical Bernoulli bond percolation on Z^2.
We prove that in the critical Bernoulli percolation on two dimensional lattice slabs the probabilities of open left-right crossings of rectangles with any given aspect ratio are uniformly positive.
We show that adding epsilon-Bernoulli percolation to an everywhere percolating subgraph of Z^2 results in a graph which has large scale geometry similar to that of supercritical Bernoulli percolation, in various specific senses. We conjecture similar behavior in higher dimensions.
We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$. - for $p>p_c$, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound $θ(p)\ge\tfrac{p-p_c}{p(1-p_c)}$. This note presents the argument of \cite{DumTas15}, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework o…
Iterated Graph Systems (II): Bernoulli percolation and critical-exponent universality classes on hierarchical lattices
研究层级晶格上的伯努利渗流，证明有限维数族不能决定普适类，并否证普适性需可公度重整化尺度的猜想。
The statistical behavior of the size (or mass) of the largest cluster in subcritical percolation on a finite lattice of size $N$ is investigated (below the upper critical dimension, presumably $d_c=6$). It is argued that as $N \to \infty$ the cumulative distribution function converges to the Fisher-Tippett (or Gumbel) distribution $e^{-e^{-z}}$ in a certain weak sense (when suitably normalized). The mean grows like $s_ξ^* \log N$, where $s_ξ^*(p)$ is a ``crossover size''. The standard deviation is bounded near $s_ξ^* π/\sqrt{6}$ with persistent fluctuations due to discreteness. These predictio…
We prove Ornstein-Zernike behaviour in every direction for finite connection functions of bond percolation on $\mathbb{Z}^{d}$ for $d\geq3$ when $p,$ the probability of occupation of a bond, is sufficiently close to $1.$ Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.
We provide a new proof of the near-critical scaling relation $β=ξ_1ν$ for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo's formula, but rather relates differences in crossing probabilities at different scales. The argument is shorter and more robust than previous ones and is more likely to be adapted to other models. The same approach may be used to prove the other scaling relations appearing in Kesten's work.
For the d-dimensional model of a subcritical bond percolation (p<p_c) and a point \vec{a} in Z^d, we prove that a cluster conditioned on connecting points (0,...,0) and n\vec{a} if scaled by 1/(n|vec{a}|) along \vec{a} and by 1/sqrt{n} in the orthogonal direction converges asymptotically to Time x (d-1)-dimensional Brownian Bridge.
We prove that for Bernoulli percolation on $\mathbb{Z}^d$, $d\geq 2$, the percolation density is an analytic function of the parameter in the supercritical interval. For this we introduce some techniques that have further implications. In particular, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that $p_c^{bond} <1/2$ for certain families of triangulations for which Benjamini \& Schramm conjectured that $p_c^{site} \leq 1/2$.
Phase transition for strongly correlated percolation models on supercritical Bernoulli clusters
本文研究超临界 Bernoulli 团簇上高斯自由场水平集与随机交织空置集的渗流相变，证明其在确定性临界水平发生非平凡相变，并发展了淬火树嵌入控制方法。
Superconcentration and chaos in Bernoulli percolation
本文研究了伯努利渗流中化学距离的超浓度现象，通过动态有效半径和晶格动物理论量化了测地线的混沌行为。
We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the …
Subdiffusive concentration for the chemical distance in Bernoulli percolation
针对超临界伯努利渗流中图距离的扩散集中性，本文建立次扩散集中不等式改进结果，受首达渗流工作启发并结合有效半径等新工具，以一步重正化处理边重采样影响，方差次线性界为其推论。
Sharp bounds on the half-space two-point function for high-dimensional Bernoulli percolation
该研究在高维伯努利渗流中推导了半空间两点函数的精确界，完善了之前的研究成果并解决了相关问题。
We prove that the heavy clusters are indistinguishable for Bernoulli percolation on quasi-transitive nonunimodular graphs. As an application, we show that the uniqueness threshold of any quasi-transitive graph is also the threshold for connectivity decay. This resolves a question of Lyons and Schramm (1999) in the Bernoulli percolation case and confirms a conjecture of Schonmann (2001). We also prove that every infinite cluster of Bernoulli percolation on a nonamenable quasi-transitive graph is transient almost surely.
An alternative approach for the mean-field behaviour of spread-out Bernoulli percolation in dimensions $d>6$
针对 $d>6$ 的充分分散 Bernoulli 渗流，论文提出推导平均场指数的新途径，并证明临界与近临界情形下全空间、半空间两点函数的上界。
In this work we study the Poisson Boolean model of percolation in locally compact Polish metric spaces and we prove the invariance of subcritical and supercritical phases under mm-quasi-isometries. In other words, we prove that if the Poisson Boolean model of percolation is subcritical or supercritical (or exhibits phase transition) in a metric space M which is mm-quasi-isometric to a metric space N, then these phases also exist for the Poisson Boolean model of percolation in N. Then we apply these results to understand the phenomenon of phase transition in a large family of metric spaces. Ind…
In high dimensional percolation at parameter $p < p_c$, the one-arm probability $π_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $π_p(n) / π_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at "mesoscopic distance" fr…
We consider boundary roughness for the ``droplet'' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least $l^2$, for large $l$. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large $l$ this maximum is at least of order $l^{1/3}(\log l)^{-2/3}$. This complements the upper bound of order $l^{1/3}(\log l)^{2/3}$ known for the average local roughness. The exponent 1/3 on $l$ here is in keeping wit…
We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $β<β_c$, and the mean-field lower bound $\mathbb{P}_β[0\longleftrightarrow\infty]\ge (β-β_c)/β$ for $β>β_c$. For finite-range models, we also prove that for any $β<β_c$, the probability of an open path from the origin to distance $n$ decays exponentially fast in $n$. For the Ising model, we pro…
We consider a multiscale Boolean percolation on $\mathbb R^d$ with radius distribution $μ$ on $[1,+\infty)$, $d\ge 2$. The model is defined by superposing the original Boolean percolation model with radius distribution $μ$ with a countable number of scaled independent copies. The $n$-th copy is a Boolean percolation with radius distribution $μ|_{[1,κ]}$ rescaled by $κ^{n}$. We prove that under some regularity assumption on $μ$, the subcritical phase of the multiscale model is sharp for $κ$ large enough. Moreover, we prove that the existence of an unbounded connected component depends only on t…
We consider the Poisson Boolean model of continuum percolation. We show that there is a subcritical phase if and only if $E(R^d)$ is finite, where $R$ denotes the radius of the balls around Poisson points and $d$ denotes the dimension. We also give related results concerning the integrability of the diameter of subcritical clusters.
Subcritical regimes in the Poisson Boolean percolation on Ahlfors regular spaces
本研究将Gouéré关于欧氏空间上Poisson Boolean渗流次临界区域存在性与半径分布矩条件等价的结论，推广至Ahlfors正则度量测度空间，拓展了渗流理论的适用范围。
We consider Bernoulli hyper-edge percolation on $\mathbb{Z}^d$. This model is a generalization of Bernoulli bond percolation. An edge connects exactly two vertices and a hyper-edge connects more than two vertices. As in the classical Bernoulli bond percolation, we open hyper-edges independently in a homogeneous manner with certain probabilities parameterized by a parameter $u\in[0,1]$. We discuss conditions for non-trivial phase transitions when $u$ varies. We discuss the conditions for the uniqueness of the infinite cluster. Also, we provide conditions under which the Grimmett-Marstrand type …
We prove that there exist natural generalizations of the classical bootstrap percolation model on $\mathbb{Z}^2$ that have non-trivial critical probabilities, and moreover we characterize all homogeneous, local, monotone models with this property. Van Enter (in the case $d=r=2$) and Schonmann (for all $d \geq r \geq 2$) proved that $r$-neighbour bootstrap percolation models have trivial critical probabilities on $\mathbb{Z}^d$ for every choice of the parameters $d \geq r \geq 2$: that is, an initial set of density $p$ almost surely percolates $\mathbb{Z}^d$ for every $p>0$. These results effec…
The goal of this note is to announce certain results in orbit equivalence theory, especially concerning the approximation of p.m.p. standard equivalence relations by increasing sequence of sub-relations, with applications to the behavior of the Bernoulli percolation on graphs at the threshold pu. Résumé en Français. Approximations de relations d'équivalence standards et percolation de Bernoulli à p\_u.
We study subcritical two-dimensional oriented percolation seen from its rightmost point on the set of infinite configurations which are bounded above. This a Feller process whose state space is not compact and has no invariant measures. We prove that it converges in distribution to a measure which charges only finite configurations.
We consider the Boolean model on $\R^d$. We prove some equivalences between subcritical percolation properties. Let us introduce some notations to state one of these equivalences. Let $C$ denote the connected component of the origin in the Boolean model. Let $|C|$ denotes its volume. Let $\ell$ denote the maximal length of a chain of random balls from the origin. Under optimal integrability conditions on the radii, we prove that $E(|C|)$ is finite if and only if there exists $A,B >0$ such that $¶(\ell \ge n) \le Ae^{-Bn}$ for all $n \ge 1$.
Sharpness of Bernoulli percolation via couplings
本文利用受 Russo 启发的随机耦合控制，在一般传递无限图上建立 Bernoulli 渗流连接概率估计，无需微分公式即证明相变锐性，并导出次临界指数衰减与超临界平均场下界。
We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.
We prove that the Poisson-Boolean percolation on $\mathbb{R}^d$ undergoes a sharp phase transition in any dimension under the assumption that the radius distribution has a $5d-3$ finite moment (in particular we do not assume that the distribution is bounded). More precisely, we prove that: -In the whole subcritical regime, the expected size of the cluster of the origin is finite, and furthermore we obtain bounds for the origin to be connected to distance $n$: when the radius distribution has a finite exponential moment, the probability decays exponentially fast in $n$, and when the radius dist…
We consider the Bernoulli Boolean discrete percolation model on the d-dimensional integer lattice. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the intensity of the underlying point process is small enough. We also study a Harris graphical procedure to construct, forward in time, particle systems with interactions of infinite range under the assumption that the corresponding generator admits a Kalikow-type decomposition. We do so by using the subcriticality of the boolean…
We apply a variation on the methods of Duminil-Copin, Raoufi, and Tassion to establish a new differential inequality applying to both Bernoulli percolation and the Fortuin-Kasteleyn random cluster model. This differential inequality has a similar form to that derived for Bernoulli percolation by Menshikov but with the important difference that it describes the distribution of the volume of a cluster rather than of its radius. We apply this differential inequality to prove the following: The critical exponent inequalities $γ\leq δ-1$ and $Δ\leq γ+1$ hold for percolation and the random cluster m…
We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $β\geq 0$. Intuitively, larger $β$ means more randomness. In particular, at $β=0$ we start with loops of length 1 and as $β$ crosses a critical value $β_c$, infinite loops start to occur almost surely. Our random loop models admit a natural comparison to bond percolation with $p=1-e^{-β}$ on the same graph to obtain a lower bound on $β_c$. For those graphs of diverging vertex degree where $β_c$ and the critical parameter for percolation have been calculated explicitly,…
In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom's North-East-Center majority rule cellular automaton.
The diffusivity of supercritical Bernoulli percolation is infinitely differentiable
本文证明超临界 $\mathbb{Z}^d$-Bernoulli 渗流的扩散系数与电导率无限可微，并通过受扰校正方程、多尺度几何重整化及新分解技术推广 Kozlov 的结果。
Subcritical percolation and network archaeology on random recursive tree substrate networks
研究随机递归树基底网络的网络考古学问题，利用辅助亚临界键渗流揭示树状重整化结构，通过在最大辅助渗流组件内应用乔丹中心性，为循环网络构建确定性大小的根置信集。
We consider some continuum percolation models. We are mainly interested in giving some sufficient conditions for absence of percolation. We give some general conditions and then focuse on two examples. The first one is a multiscale percolation model based on the Boolean model. It was introduced by Meester and Roy and subsequently studied by Menshikov, Popov and Vachkovskaia. The second one is based on the stable marriage of Poisson and Lebesgue introduced by Hoffman, Holroyd and Peres and whose percolation properties have been studied by Freire, Popov and Vachkovskaia.
We study a competition model on $\mathbb{Z}^d$ where the two infections are driven by supercritical Bernoulli percolations with distinct parameters $p$ and $q$. We prove that, for any $q$, there exist at most countably many values of $p<\min(q, \overrightarrow{p\_c})$ such that coexistence can occur.
Upper bounds for critical probabilities in Bernoulli Percolation models
本文研究了有向和无向伯努利渗流模型中临界概率的上界，针对d≥3的所有维度给出了严格的结果。
We consider the Poisson Boolean percolation model in $\mathbb{R}^2$, where the radii of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in $\mathbb{R}^d$, for any $d\ge2$, finite moment of order $d$ is both necessary and suf…
We consider Markov random fields of discrete spins on the lattice $\Zd$. We use a technique of coupling of conditional distributions. If under the coupling the disagreement cluster is "sufficiently" subcritical, then we prove the Poincaré inequality. In the whole subcritical regime, we have a weak Poincaré inequality and corresponding polynomial upper bound for the relaxation of the associated Glauber dynamics.
Non-perturbative finite connectivities in supercritical Bernoulli percolation on $\mathbb{Z}^d$ with $d\geq 3$
本文解决了超临界 Bernoulli 逾渗中有限连通概率的长期问题，通过更新结构与改进的粗粒化方法，推导出尖锐的 Ornstein--Zernike 渐近形式。
Locality of Bernoulli Site Percolation on Transitive Graphs
本文在顶点传递图上证明了伯努利点渗流临界概率的局部性，通过发展点渗流形式的双鬼不等式、锐阈值、滚雪球估计及非单模高度方法，并建立多尺度论证所需的丰富管几何。
For directed Bernoulli last passage percolation with i.i.d.~weights on vertices over a $n\times n$ grid and for $n$ large enough, the geodesics are shown to be concentrated in a cylinder, centered on the main diagonal and of width of order $n^{(2κ+2)/(2κ+3)}\sqrt{\ln n}$, where $1\leκ<\infty$ is the curvature power of the shape function at $(1,1)$. The methodology of proof is robust enough to also apply to directed Bernoulli first passage site percolation, and further to longest common subsequences in random words.
Crossover from subcritical to critical decay: random walk, self-avoiding walk, percolation
本文研究了平衡统计力学中亚临界两点函数的奥恩斯坦-泽恩基衰减，证明了在适当假设下， Ornstein-Zernike方程在Z^d上的解行为与布朗运动带漂移的格林函数乘以各向异性指数衰减因子一致，适用于多种随机游走、自避游走和渗透模型。
Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach
针对分层图上伯努利渗流的连通概率单调性猜想，提出逐层构建渗流构型的马尔可夫链方法，证明有限基图下该单调性对足够大的层差成立，为相关渗流问题提供新研究思路。
We consider Bernoulli percolation on a locally finite quasi-transitive unimodular graph and prove that two infinite clusters cannot have infinitely many pairs of vertices at distance 1 from one another or, in other words, that such graphs exhibit ``cluster repulsion.'' This partially answers a question of Häggström, Peres and Schonmann.
The Contact Process Can Survive on a Slightly Subcritical Dynamical Percolation Cluster
该研究证明在略微亚临界动态渗透簇上，接触过程可以生存，扩展了一维情况下的结果。
We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $θ(p)>0$, we prove that, for any $α>0$, there exists $κ>0$ such that, with probability larger than $1-1/n^α$, every pair of vertices inside the box $Λ(n)$ are joined by a path having at most $κ(\ln n)^2$ closed sites.
Benjamini and Kesten introduced in 1995 the problem of embedding infinite binary sequences into a Bernoulli percolation configuration, known as "percolation of words". We give a positive answer to their Open Problem 2: almost surely, all words are seen for site percolation on Z^3 with parameter p = 1/2. We also extend this result in various directions, proving the same result for any dimension d at least three and for any value p in the interval (p_c(Z^d), 1 - p_c(Z^d)), and for restrictions to slabs. Finally, we provide an explicit estimate on the probability to find all words starting from a…
We prove upper bounds on the one-arm exponent $η_1$ for a class of dependent percolation models which generalise Bernoulli percolation; while our main interest is level set percolation of Gaussian fields, the arguments apply to other models in the Bernoulli percolation universality class, including Poisson-Voronoi and Poisson-Boolean percolation. More precisely, in dimension $d=2$ we prove that $η_1 \le 1/3$ for continuous Gaussian fields with rapid correlation decay (e.g. the Bargmann-Fock field), and in $d \ge 3$ we prove $η_1 \le d/3$ for finite-range fields, both discrete and continuous, a…
We consider the Bernoulli bond percolation process $\mathbb{P}_{p,p'}$ on the nearest-neighbor edges of $\mathbb{Z}^d$, which are open independently with probability $p<p_c$, except for those lying on the first coordinate axis, for which this probability is $p'$. Define \[ξ_{p,p'}:=-\lim_{n\to\infty}n^{-1}\log \mathbb{P}_{p,p'}(0\leftrightarrow n\mathbf {e}_1)\] and $ξ_p:=ξ_{p,p}$. We show that there exists $p_c'=p_c'(p,d)$ such that $ξ_{p,p'}=ξ_p$ if $p'<p_c'$ and $ξ_{p,p'}<ξ_p$ if $p'>p_c'$. Moreover, $p_c'(p,2)=p_c'(p,3)=p$, and $p_c'(p,d)>p$ for $d\geq 4$. We also analyze the behavior of $…
Consider Bernoulli bond percolation on a locally finite, connected graph $G$ and let $p_{\mathrm{cut}}$ be the threshold corresponding to a "first-moment method" lower bound. Kahn (\textit{Electron.\ Comm.\ Probab.\ Volume 8, 184-187.} (2003)) constructed a counter-example to Lyons' conjecture of $p_{\mathrm{cut}}=p_c$ and proposed a modification. Here we give a positive answer to Kahn's modified question. The key observation is that in Kahn's modification, the new expectation quantity also appears in the differential inequality of one-arm events. This links the question to a lemma of Duminil-…
Exponential decay of the volume for Bernoulli percolation: a proof via stochastic comparison
针对连通传递等特性的无限局部有限图上的次临界伯努利渗流，提出不依赖微分不等式、受Russo相关论文启发的基于随机比较技术的团簇体积指数衰减性质新短证明。
The present paper is a continuation of \cite{dHP07b}. The object of interest is a two-dimensional model of a directed copolymer, consisting of a random concatenation of hydrophobic and hydrophilic monomers, immersed in an emulsion, consisting of large blocks of oil and water arranged in a percolation-type fashion. The copolymer interacts with the emulsion through an interaction Hamiltonian that favors matches and disfavors mismatches between the monomers and the solvents, in such a way that the interaction with the oil is stronger than with the water. The model has two regimes, supercritical a…
We prove that for Voronoi percolation on $\mathbb{R}^d$, there exists $p_c\in[0,1]$ such that - for $p<p_c$, there exists $c_p>0$ such that $\mathbb{P}_p[0\text{ connected to distance }n]\leq \exp(-c_p n)$, - there exists $c>0$ such that for $p>p_c$, $\mathbb{P}_p[0\text{ connected to }\infty]\geq c(p-p_c)$. For dimension 2, this result offers a new way of showing that $p_c(2)=1/2$. This paper belongs to a series of papers using the theory of algorithms to prove sharpness of the phase transition.
In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as p approaches p_c. Closely related is the self-destructive percolation process, i…
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