Related papers: Percolation on random 2-lifts
As a generation of the classical percolation, clique percolation focuses on the connection of cliques in a graph, where the connection of two $k$-cliques means that they share at least $l<k$ vertices. In this paper, we develop a theoretical…
The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster…
In 1999, Zhang proved that, for first passage percolation on the square lattice $\mathbb{Z}^2$ with i.i.d. non-negative edge weights, if the probability that the passage time distribution of an edge $P(t_e = 0) =1/2 $, the critical value…
In this paper, we study a model of long-range site percolation on graphs of bounded degree, namely the Boolean percolation model. In this model, each vertex of an infinite connected graph is the center of a ball of random radius, and…
Percolation in an information-theoretically secure graph is considered where both the legitimate and the eavesdropper nodes are distributed as Poisson point processes. For both the path-loss and the path-loss plus fading model, upper and…
Several results are presented for site percolation on quasi-transitive, planar graphs $G$ with one end, when properly embedded in either the Euclidean or hyperbolic plane. If $(G_1,G_2)$ is a matching pair derived from some quasi-transitive…
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…
\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is…
We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated…
In this article, we study the critical percolation threshold $p_c$ for $d$-regular graphs. It is well-known that $p_c \geq \frac{1}{d-1}$ for such graphs, with equality holding for the $d$-regular tree. We prove that among all…
The divide and color model on a graph $G$ arises by first deleting each edge of $G$ with probability $1-p$ independently of each other, then coloring the resulting connected components (\emph{i.e.}, every vertex in the component) black or…
We study Bernoulli percolations on random lattices of the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process of these random lattices…
We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two…
The aim of these notes is to give a quick introduction to FK-percolation, focusing on certain recent results about the phase transition of the two dimensional model, namely its continuity or discontinuity depending on the cluster weight…
Given a branching random walk on a graph, we consider two kinds of truncations: by inhibiting the reproduction outside a subset of vertices and by allowing at most $m$ particles per site. We investigate the convergence of weak and strong…
We study a one parameter family of random graph models that spans a continuum between traditional random graphs of the Erd\H{o}s-R\'enyi type, where there is no underlying structure, and percolation models, where the possible edges are…
Given an infinite connected graph $G$, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges of the graph, a process called first-passage percolation. Assume that the graph is infinite and of bounded degree.…
A question relating the critical probability for percolation, the critical probability for a unique infinite cluster and graph limits is presented, together with some partial results.
Percolation is the paradigm for random connectivity and has been one of the most applied statistical models. With simple geometrical rules a transition is obtained which is related to magnetic models. This transition is, in all dimensions,…
In this paper, we study the order of the largest connected component of a random graph having two sources of randomness: first, the graph is chosen randomly from all graphs with a given degree sequence, and then bond percolation is applied.…