Related papers: Directed percolation in two dimensions: An exact s…

We present an algorithm to compute the exact probability $R_{n}(p)$ for a site percolation cluster to span an $n\times n$ square lattice at occupancy $p$. The algorithm has time and space complexity $O(\lambda^n)$ with $\lambda \approx…

In high dimensional percolation at parameter $p < p_c$, the one-arm probability $\pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $\pi_p(n) / \pi_{p_c}(n)$, establishing a form…

The width W of the active region around an active moving wall in a directed percolation process diverges at the percolation threshold p_c as W \simeq A \epsilon^{-\nu_\parallel} \ln(\epsilon_0/\epsilon), with \epsilon=p_c-p, \epsilon_0 a…

We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a…

Suppose each site independently and randomly chooses some sites around it, and it is weakly (strongly) connected with them (if there choose each other). What is the probability that the weak (strong) connected cluster is infinite? We…

We simulate the bond and site percolation models on several three-dimensional lattices, including the diamond, body-centered cubic, and face-centered cubic lattices. As on the simple-cubic lattice [Phys. Rev. E, \textbf{87} 052107 (2013)],…

We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0 and |\cdot| is a…

We consider long-range percolation in dimension $d\geq 1$, where distinct sites $x$ and $y$ are connected with probability $p_{x,y}\in[0,1]$. Assuming that $p_{x,y}$ is translation invariant and that $p_{x,y}=\|x-y\|^{-s+o(1)}$ with $s>2d$,…

We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the…

We study the percolation phase transition on preferential attachment models, in which vertices enter with $m$ edges and attach proportionally to their degree plus $\delta$. We identify the critical percolation threshold as…

We investigate a model of epidemic spreading with partial immunization which is controlled by two probabilities, namely, for first infections, $p_0$, and reinfections, $p$. When the two probabilities are equal, the model reduces to directed…

We consider the fractal dimensions d_k of the k-connected part of percolation clusters in two dimensions, generalizing the cluster (k=1) and backbone (k=2) dimensions. The codimensions X_k = 2-d_k describe the asymptotic decay of the…

We provide the first nontrivial upper bound for the chemical distance exponent in two-dimensional critical percolation. Specifically, we prove that the expected length of the shortest horizontal crossing path of a box of side length $n$ in…

We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed $\beta$, we show that the…

The aim of this paper is to explore possible ways of extending Smirnov's proof of Cardy's formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we…

Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on…

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. This is a special case of a conjecture due to O.…

We use SLE(6) paths to construct a process of continuum nonsimple loops in the plane and prove that this process coincides with the full continuum scaling limit of 2D critical site percolation on the triangular lattice -- that is, the…

We divide the circular boundary of a hyperbolic lattice into four equal intervals, and study the probability of a percolation crossing between an opposite pair, as a function of the bond occupation probability p. We consider the {7,3}…

We study survival and extinction of a long-range infection process on a diluted one-dimensional lattice in discrete time. The infection can spread to distant vertices according to a Pareto distribution, however spreading is also prohibited…