Related papers: Line-of-sight percolation
The percolated random geometric graph $G_n(\lambda, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, and every pair of vertices at distance at most 1 independently forms an edge with probability $p$. For a…
The Hamming torus of dimension $d$ is the graph with vertices $\{1,\dots,n\}^d$ and an edge between any two vertices that differ in a single coordinate. Bootstrap percolation with threshold $\theta$ starts with a random set of open…
We consider a version of continuum long-range percolation on finite boxes of $\mathbb{R}^d$ in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance $r$ is connected with…
We consider the random connection model in which an edge between two Poisson points at distance $r$ is present with probability $g(r)$. We conduct an extreme value analysis on this model, namely by investigating the longest edge with at…
We study the number of clusters in two-dimensional (2d) critical percolation, N_Gamma, which intersect a given subset of bonds, Gamma. In the simplest case, when Gamma is a simple closed curve, N_Gamma is related to the entanglement entropy…
We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by Omega <= 72/91, omega = D Omega <= 3/2, and Delta_1 = nu omega <= 2, based upon Cardy's result for the critical crossing probability on an…
An upper bound for the critical probability of long range bond percolation in $d=2$ and $d=3$ is obtained by connecting the bond percolation with the SIR epidemic model, thus complementing the lower bound result in Frei and Perkins…
For a given homogeneous Poisson point process in $\mathbb{R}^d$ two points are connected by an edge if their distance is bounded by a prescribed distance parameter. The behaviour of the resulting random graph, the Gilbert graph or random…
The scaling of the tails of the probability of a system to percolate only in the horizontal direction $\pi_{hs}$ was investigated numerically for correlated site-bond percolation model for $q=1,2,3,4$.We have to demonstrate that the tails…
We consider connectivity properties and asymptotic slopes for certain random directed graphs on $Z^2$ in which the set of points $C_o$ that the origin connects to is always infinite. We obtain conditions under which the complement of $C_o$…
We consider a class of random loop models (including the random interchange process) that are parametrised by a time parameter $\beta\geq 0$. Intuitively, larger $\beta$ means more randomness. In particular, at $\beta=0$ we start with loops…
Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio…
We consider the following oriented percolation model of $\mathbb {N} \times \mathbb{Z}^d$: we equip $\mathbb {N}\times \mathbb{Z}^d$ with the edge set $\{[(n,x),(n+1,y)] | n\in \mathbb {N}, x,y\in \mathbb{Z}^d\}$, and we say that each edge…
Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in…
We show that for critical site percolation on the triangular lattice two new observables have conformally invariant scaling limits. In particular the expected number of clusters separating two pairs of points converges to an explicit…
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…
Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in $d$-space, with distance parameter $r$ and intensities $\lambda,\mu$. For any $\lambda>0$ we consider the percolation…
A Poisson outdegree-one graph is an oriented graph based on a Poisson point process such that each vertex has only one outgoing edge. The paper focuses on the absence of percolation for such graphs. Our main result is based on two…
We study the locality of critical percolation on finite graphs: let $G_n$ be a sequence of finite graphs, converging locally weakly to a (random, rooted) infinite graph $G$. Consider Bernoulli edge percolation: does the critical probability…
In this paper, we study the probability of visiting a distant point $a\in \mathbb{Z}^4$ by critical branching random walk starting from the origin. We prove that this probability is bounded by $1/(|a|^2\log |a|)$ up to a constant.