Yevgeniy Kovchegov
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.
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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.
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…
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.
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…
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 $n \ge 1$.
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 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…
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…
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.
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…
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…
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…
We consider a discrete time biased random walk conditioned to avoid Bernoulli obstacles on ${\mathbb Z}^d$ ($d\geq 2$) up to time $N$. This model is known to undergo a phase transition: for a large bias, the walk is ballistic whereas for a small bias, it is sub-ballistic. We prove that in the sub-ballistic phase, the random walk is contained in a ball of radius $O(N^{1/(d+2)})$, which is the same scale as for the unbiased case. As an intermediate step, we also prove large deviation principles for the endpoint distribution for the unbiased random walk at scales between $N^{1/(d+2)}$ and $o(N^{d…
In this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander.
We investigate the problem of embedding infinite binary sequences into Bernoulli site percolation on $\mathbb{Z}^d$ with parameter $p$, known also as percolation of words.\ In 1995, I.\ Benjamini and H.\ Kesten proved that, for $d \geq 10$ and $p=1/2$, all sequences can be embedded, almost surely. They conjectured that the same should hold for $d \geq 3$. In this paper we consider $d \geq 3$ and $p \in (p_c(d), 1-p_c(d))$, where $p_c(d)<1/2$ is the critical threshold for site percolation on $\mathbb{Z}^d$. We show that there exists an integer $M = M (p)$, such that, a.s., every binary sequence…
We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\mathbf{T} , \cdot)$, where $\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\text{th}$-order Taylor expansion of $g(\mathbf{T} , \cdot)$ at criticality in terms of limits of martingales defined from $\mathbf{T}$ (this requires a moment condition d…
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 anisotropic independent bond percolation model on the $d$-dimensional hypercubic lattice, $d\geq 2$, with parameter $p$. We show that the two point connectivity function $P_{p}(\{(0,\dots,0)\leftrightarrow (n,0,\dots,0)\})$ is a monotone function in $n$ when the parameter $p$ is close enough to 0. Analogously, we show that truncated connectivity function $P_{p}(\{(0,\dots,0)\leftrightarrow (n,0,\dots,0), (0,\dots,0)\nleftrightarrow\infty\})$ is also a monotone function in $n$ when $p$ is close to 1.
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.
The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $μ$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.
Existence of subcritical percolation phases for generalised weight-dependent random connection models
本文研究带标记泊松点过程上的广义权重依赖随机连接模型，通过控制长边给出临界环带穿越强度为正及次临界相存在的充分条件，并建立典型簇直径与规模的尾界。
We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster C\_p [11]. We are interested in the regularity properties of the chemical distance for supercritical Bernoulli percolation. The chemical distance between two points x, y $\in$ C\_p corresponds to the length of the shortest path in C\_p joining the two points. The chemical distance between 0 and nx grows asymptotically like n$μ$\_p (x). We aim to…
We generalize the standard site percolation model on the $d$-dimensional lattice to a model on random tessellations of $\mathbb R^d$. We prove the uniqueness of the infinite cluster by adapting the Burton-Keane argument \cite{burton1989density}, develop two frameworks that imply the non-triviality of the phase transition and show that large classes of random tessellations fit into one of these frameworks. Our focus is on a very general approach that goes well beyond the typical Poisson driven models. The most interesting examples might be Voronoi tessellations induced by determinantal processe…
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。
Fluctuations around the diagonal in Bernoulli-Exponential first passage percolation
本文研究 Bernoulli-Exponential 首通渗流中团簇边界在对角线附近的涨落，证明其重标度单点涨落收敛于以渗流水平 s 为索引的新分布族，该族在 s=0 时为高斯分布，并随 s 增大趋于 Tracy--Widom 分布。
The probability of connection between two vertices cannot be monotone with the distance for Bernoulli Percolation on transitive graphs
本文研究了伯努利渗流中两个顶点连接概率与距离的关系，发现其不必然随距离单调递减，并定义了临界点现象。
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.
A finite-graph conjecture related to $θ(p_c)=0$ for Bernoulli bond percolation on $\mathbb Z^d$
该研究提出有限图伯努利键渗流的一个猜想，可推出d≥2时Z^d上θ(p_c)=0，证明有限平面图情形成立，反例显示连通性结构不可或缺。
We consider the standard model of i.i.d. first passage percolation on $\mathbb{Z}^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty]) < p_c(d)$, it is known that the time constant $μ_G$ exists. We are interested in the regularity properties of the map $G\mapstoμ_G$. We first study the specific case of distributions of the form $G_p=pδ_1+(1-p)δ_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We show that the function $p\mapsto …
We consider Bernoulli first-passage percolation on the $d$-dimensional hypercubic lattice with $d \geq 2$. The passage time of edge $e$ is $0$ with probability $p$ and $1$ with probability $1-p$, independently of each other. Let $p_c$ be the critical probability for percolation of edges with passage time $0$. When $0\leq p<p_c$, there exists a nonrandom, nonempty compact convex set $\mathcal{B}_p$ such that the set of vertices to which the first-passage time from the origin is within $t$ is well-approximated by $t\mathcal{B}_p$ for all large $t$, with probability one. The aim of this paper is …
Subcritical Boolean percolation on graphs of bounded degree
本文研究了图 bounded degree 上的亚临界布尔渗流模型，提出了半径分布的充分条件以确保亚临界相的存在，并探讨了连通分量大小的指数衰减条件。
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.
We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter $1-ε$. These passage times induce a random pseudo-metric $T_ε$ on $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_ε(0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $μ_ε(z)$ called the time constant. We investiga…
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.
Enlargement of subgraphs of infinite graphs by Bernoulli percolation
本文研究无限图子图在加入伯努利渗流开放边后性质的变化，定义两类临界概率并与哈马米临界概率比较，探讨子图的瞬态性、割点数量、复发性及连通性等属性。
We establish several equivalent characterisations of the anchored isoperimetric dimension of supercritical clusters in Bernoulli bond percolation on transitive graphs. We deduce from these characterisations together with a theorem of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin that if $G$ is a transient transitive graph then the infinite clusters of Bernoulli percolation on $G$ are transient for $p$ sufficiently close to $1$. It remains open to extend this result down to the critical probability. Along the way we establish two new cluster repulsion inequalities that are of independent in…
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.
In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitively inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different "types" of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponentially. Our proof extends the proof of sharpness of the phase transition for homogeneous percolation on vertex-transitive g…
We introduce a percolation model on $\mathbb{Z}^d$, $d \geq 3$, in which the discrete lines of vertices that are parallel to the coordinate axis are entirely removed at random and independently of each other. In this way a vertex belongs to the vacant set $\mathcal{V}$ if and only if none of the $d$ lines to which it belongs, is removed. We show the existence of a phase transition for $\mathcal{V}$ as the probability of removing the lines is varied. We also establish that, in the certain region of parameters space where $\mathcal{V}$ contains an infinite component, the truncated connectivity f…
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.
Largest component and sharpness in continuum percolation
本文研究了在非临界情况下泊松随机连接模型中大连接组件的行为，证明了在亚临界情况下最大组件大小在窗口体积上以对数增长，在超临界情况下线性增长，并证明了在亚临界情况下原点集群的尾部呈指数衰减。
The aim of this paper is to generalize the well-known asymptotic shape result for first-passage percolation on $\Zd$ to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet points to a deterministic shape that does not depend on the random environment. As a special case of the previous result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result. Some various examples are also given…
Fix a graph $G$ in which every edge is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ are CA-connected if $u$ and $v$ may be connected using any subset of $k - 1$ colors. CA-connectivity is an equivalence relation dividing the vertex set into classes called CA-components. In two recent papers, Ráth, Varga, Fekete, and Molontay, and Lichev and Schapira studied the size of the largest CA-component in a randomly colored random graph. The second of these works distinguished and studied three regimes (supercritical, intermediate, and subcritical) in which the largest CA-component has …
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…
We show that a superposition of an $\varepsilon$-Bernoulli bond percolation and any everywhere percolating subgraph of $\mathbb Z^d$, $d\ge 2$, results in a connected subgraph, which after a renormalization dominates supercritical Bernoulli percolation. This result, which confirms a conjecture of the first author together with Häggström and Schramm (2000), is mainly motivated by obtaining finite volume characterizations of uniqueness for general percolation processes.
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