Related papers: Critical percolation of virtually free groups and …
We consider the model of random trees introduced by Devroye (1999), the so-called random split trees. The model encompasses many important randomized algorithms and data structures. We then perform supercritical Bernoulli bond-percolation…
Consider a uniformly random regular graph of a fixed degree $d\ge3$, with $n$ vertices. Suppose that each edge is open (closed), with probability $p(q=1-p)$, respectively. In 2004 Alon, Benjamini and Stacey proved that $p^*=(d-1)^{-1}$ is…
The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on…
We study the bond percolation problem in random graphs of $N$ weighted vertices, where each vertex $i$ has a prescribed weight $P_i$ and an edge can connect vertices $i$ and $j$ with rate $P_iP_j$. The problem is solved by the $q\to 1$…
We describe a probabilistic methodology, based on random walk estimates, to obtain exponential upper bounds for the probability of observing unusually small maximal components in two classical (near-)critical random graph models. More…
Consider subcritical Bernoulli bond percolation with fixed parameter p<p_c. We define a dependent site percolation model by the following procedure: for each bond cluster, we colour all vertices in the cluster black with probability r and…
We study Poisson--Voronoi percolation and its discrete analogue Bernoulli--Voronoi percolation in spaces with a non-amenable product structure. We develop a new method of proving smallness of the uniqueness threshold $p_u(\lambda)$ at small…
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…
Bootstrap percolation is a well-known activation process in a graph, in which a node becomes active when it has at least $r$ active neighbors. Such process, originally studied on regular structures, has been recently investigated also in…
Let $G$ be an infinite, connected, locally finite planar graph and consider i.i.d.\ Bernoulli$(p)$ site percolation. Write $p_c^{\mathrm{site}}(G)$ and $p_u^{\mathrm{site}}(G)$ for the critical and uniqueness thresholds. Using a…
We study Bernoulli first-passage percolation (FPP) on the triangular lattice $\mathbb{T}$ in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. Denote by $\mathcal {C}_{\infty}$ the infinite cluster with…
Let $G=(V,E)$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. In recent work, we conjectured that if $G$ is nonamenable then the matrix of critical connection probabilities…
We consider the simple random walk on the infinite cluster of the Bernoulli bond percolation of trees, and investigate the relation between the speed of the simple random walk and the retaining probability p by studying three classes of…
In r-neighbour bootstrap percolation on a graph G, a set of initially infected vertices A \subset V(G) is chosen independently at random, with density p, and new vertices are subsequently infected if they have at least r infected…
Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…
We study bootstrap percolation with the threshold parameter $\theta \geq 2$ and the initial probability $p$ on infinite periodic trees that are defined as follows. Each node of a tree has degree selected from a finite predefined set of…
Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a previous work by the second author, the proof relies on…
We define an inhomogeneous percolation model on "ladder graphs" obtained as direct products of an arbitrary graph $G = (V,E)$ and the set of integers $\mathbb{Z}$ (vertices are thought of as having a "vertical" component indexed by an…
This paper deals with the construction of a correlation decay tree (hypertree) for interacting systems modeled using graphs (hypergraphs) that can be used to compute the marginal probability of any vertex of interest. Local message passing…
We prove two results concerning percolation on general graphs. - We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies $p_c<1$, then the number of minimal cutsets of size…