Jean-Baptiste Gouéré
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.
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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.
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 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 $n \ge 1$.
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 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 percolation in a multiscale Boolean model. This model is defined as the union of scaled independent copies of a given Boolean model. The scale factor of the $n^{\textrm{th}}$ copy is $ρ^{-n}$. We prove, under optimal integrability assumptions, that no percolation occurs in the multiscale Boolean model for large enough $ρ$ if the rate of the Boolean model is below some critical value.
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 investigate percolation in the Boolean model with convex grains in high dimension. For each dimension d, one fixes a compact, convex and symmetric set K $\subset$ R d with non empty interior. In a first setting, the Boolean model is a reunion of translates of K. In a second setting, the Boolean model is a reunion of translates of K or $ρ$K for a further parameter $ρ$ $\in$ (1, 2). We give the asymptotic behavior of the percolation probability and of the percolation threshold in the two settings.
We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. 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 retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
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 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…
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。
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。
It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.
We consider Poisson Boolean percolation on $\mathbb R^d$ with power-law distribution on the radius with a finite $d$-moment for $d\ge 2$. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin--Raoufi--Tassion where subcritical sharpness is proved under the assumption that the radii distribution has a $5d-3$ finite moment. Our proofs techniques are different from their paper: we do not use randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlyin…
Widths of crossings in Poisson Boolean percolation
研究平面 Poisson Boolean 渗流模型中 $n\times n$ 正方形交叉的宽度问题，发现其宽度在临界、次临界与超临界状态下表现不同，且占据集与空置集存在差异。
Cluster sizes in subcritical soft Boolean models
研究软布尔模型中亚临界渗流相典型最大连通分量的欧几里得直径和点数尾行为，基于边权和半径尾指数给出精确判据。
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.
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。
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.
We establish the mean-field bounds $γ\ge 1$, $δ\ge 2$ and $\triangle \ge 2$ on the critical exponents of the Poisson-Boolean continuum percolation model under a moment condition on the radii; these were previously known only in the special case of fixed radii (in the case of $γ$), or not at all (in the case of $δ$ and $\triangle$). We deduce these as consequences of the mean-field bound $β\le 1$, recently established by Duminil-Copin, Raoufi and Tassion under the same moment condition, using a relative entropy method introduced by the authors in previous work.
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 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…
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 …
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。
Boolean percolation on digraphs and random exchange processes
本文在一般图论框架下研究 Lamperti 引入的长程渗流模型，探讨其与随机交换过程的联系，并阐明了格 $\mathbb{N}_0^n$ 和 $\mathbb{Z}^n$ 被本质覆盖的条件，同时证明无法覆盖有向 $n$ 叉树。
We study bond percolation on the hypercube $\{0,1\}^m$ in the slightly subcritical regime where $p = p_c (1-\varepsilon_m)$ and $\varepsilon_m = o(1)$ but $\varepsilon_m \gg 2^{-m/3}$ and study the clusters of largest volume and diameter. We establish that with high probability the largest component has cardinality $Θ\left(\varepsilon_m^{-2} \log(\varepsilon_m^3 2^m)\right)$, that the maximal diameter of all clusters is $(1+o(1)) \varepsilon_m^{-1} \log(\varepsilon_m^3 2^m)$, and that the maximal mixing time of all clusters is $Θ\left(\varepsilon_m^{-3} \log^2(\varepsilon_m^3 2^m)\right)$. The…
Continuum Percolation in a Nonstabilizing Environment
本文研究非稳定环境中基于Cox点过程的布尔模型的连续渗流相变，通过离散化与随机拉伸晶格比较，证明参数单独或联合变化时的相变存在性。
Kesten showed the exponential decay of percolation probability in the subcritical phase for the two-dimensional percolation model. This result implies his celebrated computation that $p_c=0.5$ for bond percolation in the square lattice, and site percolation in the triangular lattice, respectively. In this paper, we present a simpler proof for Kesten's theorem.
Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
本文研究强相关高斯场次临界游走集的连接概率衰减，给出不同协方差正则变化指数下的对数渐近速率，推广了高斯自由场结果，并支持关联长度指数预测。
Large deviations for subcritical bootstrap percolation on the random graph
针对Erdős-Rényi随机图上的亚临界bootstrap渗流，本文计算了小规模初始感染集感染异常多顶点事件的速率函数，确定了对应大偏差的最小代价轨迹。
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 …
Only long edges can erase the subcritical annulus-crossing phase of weight-dependent random connection models
本文针对权重依赖随机连接模型的环带穿越次临界相问题，提出更直接且适用于经典模型的长边判据，弥补既有判据在部分情形下不确定的不足。
We consider a non trivial Boolean model $Σ$ on ${\mathbb R}^d$ for $d\geq 2$. For every $x,y \in {\mathbb R}^d$ we define $T(x,y)$ as the minimum time needed to travel from $x$ to $y$ by a traveler that walks at speed $1$ outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily shows that $T(0,x)$ behaves like $μ\|x\|$ when $\|x\|$ goes to infinity, where $μ$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the positivity of $μ$. More precisely, under an almost optimal moment as…
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…
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…
Recently the study of noise sensitivity and noise stability of Boolean functions has received considerable attention. The purpose of this paper is to extend these notions in a natural way to a different class of perturbations, namely those arising from running the symmetric exclusion process for a short amount of time. In this study, the case of monotone Boolean functions will turn out to be of particular interest. We show that for this class of functions, ordinary noise sensitivity and noise sensitivity with respect to the complete graph exclusion process are equivalent. We also show this equ…
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…
Non-equilibrium coagulation processes and subcritical percolation on evolving networks
本文研究了生长网络中渗透现象，揭示了亚临界区域的临界行为及长程依赖性，发展了随机逼近等一般工具用于证明相关猜想。
Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds impr…
Detection, coverage and percolation in dynamic Boolean models with random radii based on $α$-stable processes
本文研究了基于α-稳定过程的动态布尔模型中检测、覆盖和渗流现象的尾部行为，揭示了稳定性指数和随机半径对衰减率的影响。
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.
We consider a dilute lattice obtained from the usual $\mathbb{Z}^3$ lattice by removing independently each of its columns with probability $1-ρ$. In the remaining dilute lattice independent Bernoulli bond percolation with parameter $p$ is performed. Let $ρ\mapsto p_c(ρ)$ be the critical curve which divides the subcritical and supercritical phases. We study the behavior of this curve near the disconnection threshold $ρ_c = p_c^{\text{site}}(\mathbb{Z}^2)$ and prove that, uniformly over $ρ$ it remains strictly below $1/2$ (the critical point for bond percolation on the square lattice $\mathbb{Z}…
We study monotone cellular automata (also known as $\mathcal{U}$-bootstrap percolation) in $\mathbb{Z}^d$ with random initial configurations. Confirming a conjecture of Balister, Bollobás, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non-zero for all subcritical models.
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。
We consider percolation properties of the Boolean model generated by a Gibbs point process and balls with deterministic radius. We show that for a large class of Gibbs point processes there exists a critical activity, such that percolation occurs a.s. above criticality. For locally stable Gibbs point processes we show a converse result, i.e. they do not percolate a.s. at low activity.
Starting with a percolation model in $\Z^d$ in the subcritical regime, we consider a random walk described as follows: the probability of transition from $x$ to $y$ is proportional to some function $f$ of the size of the cluster of $y$. This function is supposed to be increasing, so that the random walk is attracted by bigger clusters. For $f(t)=e^{βt}$ we prove that there is a phase transition in $β$, i.e., the random walk is subdiffusive for large $β$ and is diffusive for small $β$.
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。
We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a mean-field lower bound for the infinite cluster density in the supercritical regime. This generalizes and simplifies an earlier result of Van den Berg [Ann. App. Prob., 2011], who proved a sharp percolation phase transition on $\mathbb{Z}^2$. Our proof relies on the OSSS inequality for Boolean functions and is inspired by a series of papers by Duminil-Copin, Raou…
In the bond percolation model on a lattice, we colour vertices with $n_c$ colours independently at random according to Bernoulli distributions. A vertex can receive multiple colours and each of these colours is individually observable. The colours colour the entire component into which they fall. Our goal is to estimate the $n_c +1$ parameters of the model: the probabilities of colouring of single vertices and the probability with which an edge is open. The input data is the configuration of colours once the complete components have been coloured, without the information which vertices were or…
In this work we consider the two-dimensional percolation model arising from the majority dynamics process at a given time $t\in\mathbb{R}_+$. We show the emergence of a sharp threshold phenomenon for the box crossing event at the critical probability parameter $p_c(t)$ with polynomial size window. We then use this result in order to obtain stretched-exponential bounds on the one-arm event probability in the subcritical phase. Our results are based on differential inequalities derived from the OSSS inequality, inspired by the recent developments by Ahlberg, Broman, Griffiths, and Morris and by …
For a given dimension d $\ge$ 2 and a finite measure $ν$ on (0, +$\infty$), we consider $ξ$ a Poisson point process on R d x (0, +$\infty$) with intensity measure dc $\otimes$ $ν$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Σ$ = $\cup$ (c,r)$\in$$ξ$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $\in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Σ$ and at infinite speed inside $Σ$. By a standard application of Kingman sub-additive theorem, one easily…
We are interested in phase transitions in certain percolation models on point processes and their dependence on clustering properties of the point processes. We show that point processes with smaller void probabilities and factorial moment measures than the stationary Poisson point process exhibit non-trivial phase transition in the percolation of some coverage models based on level-sets of additive functionals of the point process. Examples of such point processes are determinantal point processes, some perturbed lattices, and more generally, negatively associated point processes. Examples of…
Bootstrap percolation is a wide class of monotone cellular automata with random initial state. In this work we develop tools for studying in full generality one of the three `universality' classes of bootstrap percolation models in two dimensions, termed subcritical. We introduce the new notion of `critical densities' serving the role of `difficulties' for critical models, but adapted to subcritical ones. We characterise the critical probability in terms of these quantities and successfully apply this link to prove new and old results for concrete models such as DTBP and Spiral as well as a ge…
Consider a $p$-random subset $A$ of initially infected vertices in the discrete cube $[L]^3$, and assume that the neighbourhood of each vertex consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each $i \in \{1,2,3\}$, where $a_1\le a_2\le a_3$. Suppose we infect any healthy vertex $x\in [L]^3$ already having $a_3+1$ infected neighbours, and that infected sites remain infected forever. In this paper we determine the critical length for percolation up to a constant factor in the exponent, for all triples $(a_1,a_2,a_3)$. To do so, we introduce a new algorithm called the bea…
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