The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d \ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${\mathbb Z}^d$ with parameter $p\in (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.
Bootstrap percolation on a graph is a deterministic process that iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure with a low density $p$. Our main focus is on this process on the product graph $\mathbb{Z}^2\times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $θ$ exhibits a sharp phase transition, while odd $θ$ yields a gradual percolation tran…
In the polluted bootstrap percolation model, vertices of the cubic lattice $\mathbb{Z}^3$ are independently declared initially occupied with probability $p$ or closed with probability $q$. Under the standard (respectively, modified) bootstrap rule, a vertex becomes occupied at a subsequent step if it is not closed and it has at least $3$ occupied neighbors (respectively, an occupied neighbor in each coordinate). We study the final density of occupied vertices as $p,q\to 0$. We show that this density converges to $1$ if $q \ll p^3(\log p^{-1})^{-3}$ for both standard and modified rules. Our pri…
Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $θ$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning occurs if every point eventually becomes occupied. The main question concerns the critical probability, that is, the minimal initial density that makes spanning likely. The graphs we consider are products of cycles of $m$ points and complete graphs of $n$ points. The major part of the paper focuses on the case when two factors are complete graphs and one factor i…
Let $T$ be a regular rooted tree. For every natural number $n$, let $B_n$ be the finite subtree of vertices with graph distance at most $n$ from the root. Consider the following forest-fire model on $B_n$: Each vertex can be "vacant" or "occupied". At time $0$ all vertices are vacant. Then the process is governed by two opposing mechanisms: Vertices become occupied at rate $1$, independently for all vertices. Independently thereof and independently for all vertices, "lightning" hits vertices at rate $λ(n) > 0$. When a vertex is hit by lightning, its occupied cluster instantaneously becomes vac…
We introduce a two-type first passage percolation competition model on infinite connected graphs as follows. Type 1 spreads through the edges of the graph at rate 1 from a single distinguished site, while all other sites are initially vacant. Once a site is occupied by type 1, it converts to type 2 at rate $ρ>0$. Sites occupied by type 2 then spread at rate $λ>0$ through vacant sites \emph{and} sites occupied by type 1, whereas type 1 can only spread through vacant sites. If the set of sites occupied by type 1 is non-empty at all times, we say type 1 \emph{survives}. In the case of a regular $…
Consider random sequential adsorption on a chequerboard lattice with arrivals at rate $1$ on light squares and at rate $λ$ on dark squares. Ultimately, each square is either occupied, or blocked by an occupied neighbour. Colour the occupied dark squares and blocked light sites {\em black}, and the remaining squares {\em white}. Independently at each meeting-point of four squares, allow diagonal connections between black squares with probability $p$; otherwise allow diagonal connections between white squares. We show that there is a critical surface of pairs $(λ, p)$, containing the pair $(1,0.…
Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle g…
We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $λ_c$. Moreover, - For $λ< λ_c$, the vacant set …
A few years ago two of us introduced, motivated by the study of certain forest-fireprocesses, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and delta is generated in three steps: First we generate a typical configuration for the ordinary percolation model with parameter p. Next, we make all sites in the infinite occupied cluster vacant. Finally, each site that was already vacant in the beginning or made vacant by the above action, becomes occupied with probability delta (independent of the other sites). Let theta(…
Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following "anisotropic" bootstrap percolation model: the neighborhood of a point (m,n) is the set \[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\] At time 0, sites are occupied with probability p. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof …
In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z^d of dimension d>=3 with threshold r=2, we prove that the final density of occupied sites converges to 1 as p and q both approach 0, regardless of their relative scaling. Our result partially resolves a conjecture of Morris, and contrasts with the d=2 case, where Gravner and McDonald proved that the cri…
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system linge…
We consider a continuum percolation model on $\R^d$, $d\geq 1$.For $t,λ\in (0,\infty)$ and $d\in\{1,2,3\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $λ\textgreater{}0$.When $d\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\geq 2$, there is a non-trivial percolation transitionin $t$, provided $λ$ and $r$ are chosen properly.The last statement means tha…
We consider continuous-time random interlacements on Z^d, d greater or equal to 3, and investigate the percolation model where a site x of Z^d is occupied if the total amount of time spent at x by all the trajectories of the interlacement at level u > 0 exceeds some given non-negative parameter, and empty otherwise. Thus, the set of occupied sites forms a subset of the interlacement at level u. We also investigate percolation properties of empty sites. A recent isomorphism theorem arXiv:1111.4818 of Sznitman enables us to "translate" some of the relevant questions into the language of level-se…
The self-destructive percolation model is defined as follows: Consider percolation with parameter $p > p_c$. Remove the infinite occupied cluster. Finally, give each vertex (or, for bond percolation, each edge) that at this stage is vacant, an extra chance $δ$ to become occupied. Let $δ_c(p)$ be the minimal value of $δ$, needed to obtain an infinite occupied cluster in the final configuration. This model was introduced some years ago by van den Berg and Brouwer. They showed that, for the site model on the square lattice (and a few other 2D lattices satisfying a special technical condition) tha…
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.
Tile \(\mathbb{R}^2\) into disjoint unit squares \(\{S_k\}_{k \geq 0}\) with the origin being the centre of \(S_0\) and say that \(S_i\) and \(S_j\) are star-adjacent if they share a corner and plus-adjacent if they share an edge. Every square is either vacant or occupied. If the occupied plus-connected component \(C^+(0)\) containing the origin is finite, it is known that the outermost boundary \(\partial^+_0\) of \(C^+(0)\) is a unique cycle surrounding the origin. For the finite occupied star-connected component \(C(0)\) containing the origin, we prove in this paper that the outermost bound…
We consider independent and $m$-dependent two-dimensional oriented site percolation with open-site density close to one started from Bernoulli product measures. We show that the probability of an occupied interval in the former process admits a lower bound which converges exponentially fast in time to the probability that the interval percolates. To this end, we derive sharp exponential bounds regarding the density of thinnings of the infinite cluster in this process started from the origin. Our approach offers a unified manner for deriving improvements to certain asymptotics invoked as auxili…
We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of Z^2 are independently occupied with small probability p, otherwise empty. Subsequently, an empty site becomes active by contact with 2 or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability exp[alpha(p)/p], where alpha(p) is between -pi^2/9 + c sqrt p and pi^2/9 + C sqrt p (-log p)^3. This corrects previo…
Bootstrap percolation on an arbitrary graph has a random initial configuration, where each vertex is occupied with probability p, independently of each other, and a deterministic spreading rule with a fixed parameter k: if a vacant site has at least k occupied neighbors at a certain time step, then it becomes occupied in the next step. This process is well-studied on Z^d; here we investigate it on regular and general infinite trees and on non-amenable Cayley graphs. The critical probability is the infimum of those values of p for which the process achieves complete occupation with positive pro…
Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate $p$, and predator particles spread only to neighboring sites occupied by prey particles at rate $1$, killing the prey particle that existed at that site. It was found that the prey can survive with non-zero probability, if $p>p_c$ with $p_c<1$. Using Monte Carlo simulations on the square lattice, we estimate the value of $p_c = 0.49…
The Russo-Seymour-Welsh Theorem for Z^2 bond or T (triangular lattice) site percolation states that at criticality, for all fixed real λ, the probability of the existence of a horizontal occupied crossing of each rectangle with size n*λn is not degenerated when n tends to infinity. Turning to site percolation on Z^2, where the self duality does not hold anymore, we prove that the analogue statement of the RSW Theorem will still be true in this case. The proof uses appropriate finite size criteria and a result of Kesten which allows us to extend existing crossings without losing too much probab…
We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the half-line left of A to the half-line right of B. We show that the probability that R has a site at distance smaller than m from AB is of order (log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.
Percolation on the stationary distributions of the voter model with stirring
Bernoulli Institute, University of Groningen（格罗宁根大学伯努利研究所）；Groningen Cognitive System and Materials Center, University of Groningen（格罗宁根大学认知系统与材料中心）；Institute of Mathematics, Statistics and Computer Science, University of São Paulo, Brazil（圣保罗大学数学、统计与计算机科学研究所）；CNRS and Sorbonne Université, Paris, France（法国国家科学研究中心与索邦大学）；Department of Statistics, University of Warwick（华威大学统计学系）
We consider a forest-fire model which, somewhat informally, is described as follows: Each site (vertex) of the square lattice is either vacant or occupied by a tree.Vacant sites become occupied at rate 1. Further, each site is hit by lightningat rate lambda. This lightning instantaneously destroys (makes vacant) the occupied cluster of the site. This model is closely related to the Drossel-Schwabl forest-fire model, which has received much attention in the physics literature. The most interesting behaviour seems to occur when the lightning rate goes to zero. In the physics literature it is bel…
We introduce a new class of two-dimensional cellular automata with a bootstrap percolation-like dynamics. Each site can be either empty or occupied by a single particle and the dynamics follows a deterministic updating rule at discrete times which allows only emptying sites. We prove that the threshold density $ρ_c$ for convergence to a completely empty configuration is non trivial, $0<ρ_c<1$, contrary to standard bootstrap percolation. Furthermore we prove that in the subcritical regime, $ρ<ρ_c$, emptying always occurs exponentially fast and that $ρ_c$ coincides with the critical density for …
