相关论文: Strict inequalities of critical probabilities on G…

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple)…

Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…

We consider supercritical bond percolation on a family of high-girth $d$-regular expanders. Alon, Benjamini and Stacey (2004) established that its critical probability for the appearance of a linear-sized ("giant'') component is…

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…

We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…

We consider the standard model of i.i.d. bond percolation on $\mathbb Z^d$ of parameter $p$. When $p>p_c$, there exists almost surely a unique infinite cluster $\mathcal C_p$. Using the recent techniques of Cerf and Dembin, we prove that…

In this paper we extend our previous results on the connectivity functions and pressure of the Random Cluster Model in the highly subcritical phase and in the highly supercritical phase, originally proved only on the cubic lattice $\Z^d$,…

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed…

We propose an approach to calculate the critical percolation threshold for finite-sized Erdos-Renyi digraphs using minimal Hamiltonian cycles. We obtain an analytically exact result, valid non-asymptotically for all graph sizes, which…

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…

We derive three critical exponents for Bernoulli site percolation on the on the Uniform Infinite Planar Triangulation (UIPT). First we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the…

We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study…

On a large finite connected graph let edges $e$ become "open" at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around…

We show that a locally finite, connected graph has a coarse embedding into a Hilbert space if and only if there exist bond percolations with arbitrarily large marginals and two-point function vanishing at infinity. We further show that the…

We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying…

In this article, we first extend the construction of random interlacements, introduced by A.S. Sznitman in [arXiv:0704.2560], to the more general setting of transient weighted graphs. We prove the Harris-FKG inequality for this model and…

We consider a dilute lattice obtained from the usual $\mathbb{Z}^3$ lattice by removing independently each of its columns with probability $1-\rho$. In the remaining dilute lattice independent Bernoulli bond percolation with parameter $p$…

Recently, the number of non-standard percolation models has proliferated. In all these models, there exists a phase transition at which long range connectivity is established, if local connectedness increases through a threshold $p_c$. In…

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the…

On a locally finite, infinite tree $T$, let $p_c(T)$ denote the critical probability for Bernoulli percolation. We prove that every positively associated, finite-range dependent percolation model on $T$ with marginals $p > p_c(T)$ must…