相关论文: Percolation of finite clusters and infinite surfac…

We present a fine-grained approach to identify clusters and perform percolation analysis in a 2D lattice system. In our approach, we develop an algorithm based on the linked-list data structure whereby the members of a cluster are nodes of…

The problem of percolation along sites of square lattice is studied. The number of contours being external boundaries for finite clusters has been estimated using geometric considerations. This estimation makes it possible to determine more…

We consider the model of a directed polymer in a random environment defined on the infinite cluster of supercritical Bernoulli bond percolation in dimensions $d \geq 3$. For this model, it was proved in arXiv:2205.06206 that for almost…

We review some of the recent progress on the scaling limit of two-dimensional critical percolation; in particular, the convergence of the exploration path to chordal SLE(6) and the "full" scaling limit of cluster interface loops. The…

We analyze the geometry of scaling limits of near-critical 2D percolation, i.e., for $p=p_c+\lambda\delta^{1/\nu}$, with $\nu=4/3$, as the lattice spacing $\delta \to 0$. Our proposed framework extends previous analyses for $p=p_c$, based…

We study long-range Bernoulli percolation on $\mathbb{Z}^d$ in which each two vertices $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta \|x-y\|^{-d-\alpha})$. It is a theorem of Noam Berger (CMP, 2002) that if…

In site percolation, vertices (sites) of a graph are open with probability p, and there is critical p, for which open vertices form an open path the long way across a graph, so a vertex at the origin is a part of an infinite connected open…

The k-neighbor graph is a directed percolation model on the hypercubic lattice Z d in which each vertex independently picks exactly k of its 2d nearest neighbors at random, and we open directed edges towards those. We prove that the…

We consider the random walk loop soup on the discrete half-plane and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to 1/2. The absence of…

We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the…

We calculate exact analytic expressions for the average cluster numbers $\langle k \rangle_{\Lambda_s}$ on infinite-length strips $\Lambda_s$, with various widths, of several different lattices, as functions of the bond occupation…

This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d$ and, more generally, on transitive graphs. For iid percolation on ${\mathbb{Z}}^d$,…

A simple, discrete, parametric model is proposed to describe conditional (correlated) deposition of particles on a surface and formation of a connecting (percolating) cluster. The surface changes spontaneously its properties (phase…

We describe the component sizes in critical independent p-bond percolation on a random d-regular graph on n vertices, where d \geq 3 is fixed and n grows. We prove mean-field behavior around the critical probability p_c=1/(d-1). In…

Aldous introduced a modification of the bond percolation process on the binary tree where clusters stop growing (freeze) as soon as they become infinite. We investigate the site version of this process on the triangular lattice where…

This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents eta and delta, for the nearest-neighbour model in very high…

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…

It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of…

We study the isoperimetric subgraphs of the infinite cluster $\textbf{C}_\infty$ for supercritical bond percolation on $\mathbb{Z}^d$ with $d\geq 3$. Specifically, we consider the subgraphs of $\textbf{C}_\infty \cap [-n,n]^d$ which have…

We present a comprehensive and versatile theoretical framework to study site and bond percolation on clustered and correlated random graphs. Our contribution can be summarized in three main points. (i) We introduce a set of iterative…