Percolation and contact processes with low-dimensional inhomogeneity
type: Item
instance of: scholarly article
title: Percolation and contact processes with low-dimensional inhomogeneity
zbMATH Open document ID: 0901.60074
author: Charles M. Newman
author: C. Chris Wu
published in: The Annals of Probability
publication date: 1998-11-24T00:00:00Z
review text: The inhomogeneous percolation model on \(\mathbb{Z}^d\), \(d\geq 2\), in consideration is defined as follows. Each bond in \(\{0\}^{d-s} \times \mathbb{Z}^s\) \((1\leq s\leq d-1)\) is open or closed with probability \(p_1\) or \(1-p_1\), respectively, and each remaining bond is open or closed with probability \(p_2\) or \(1-p_2\), respectively; all bonds of \(\mathbb{Z}^d\) behave independently. Let \(p_c (\mathbb{Z}^d)\) denote the critical value of \(p\) for the standard homogeneous (i.e. with \(p_1= p_2=p)\) percolation model on \(\mathbb{Z}^d\). The investigated problem is the appearance of percolation for the inhomogeneous model, with \(p_2= p_c (\mathbb{Z}^d)\) fixed, in dependence on \(p_1\). The relation of the studied problem to the study of the problem to show that there is no percolation at \(p=p_c (\mathbb{Z}^d)\), in the homogeneous problem, is explained. In fact, informations on the critical value \(p^s_c\) of \(p_1\) are deduced under assumptions stronger than that there is no percolation at \(p_c (\mathbb{Z}^d)\) for the homogeneous model. Related results for some inhomogeneous contact processes are presented. As a corollary, it is proved that (a) the percolation does not occur if \(s=1\), \(p_1\in[0,1)\), and \(d\) is sufficiently large; (b) there is a critical value \(p^s_c \in(p_c (\mathbb{Z}^d), p_c (\mathbb{Z}^s))\) for \(p_1\) if \(d\) is sufficiently large and \(2\leq s\leq d-3\). A related corollary for the contact processes is deduced under the assumption that an infrared bound holds for homogeneous contact processes if \(d\) is sufficiently large.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 1130488
zbMATH Keywords: percolation
zbMATH Keywords: contact process
zbMATH Keywords: inhomogeneity
reviewed by: Petr Holický
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1023481113
OpenAlex ID: W2020245218
cites work: Tree graph inequalities and critical behavior in percolation models
cites work: Discontinuity of the percolation density in one dimensional \(1/| x- y| ^ 2\) percolation models
cites work: Percolation in half-spaces: Equality of critical densities and continuity of the percolation probability
cites work: Critical exponents for the contact process under the triangle condition
cites work: The critical contact process dies out
cites work: Exponential decay for subcritical contact and percolation processes
cites work: Q4693063
cites work: The supercritical phase of percolation is well behaved
cites work: Q5753352
cites work: Mean-field critical behaviour for percolation in high dimensions
cites work: Q5734790
cites work: The critical probability of bond percolation on the square lattice equals 1/2
cites work: Q3672870
cites work: Multiple transition points for the contact process on the binary tree
cites work: On the critical behavior of the contact process in deterministic inhomogeneous environments
cites work: The critical contact process on a homogeneous tree
cites work: Triangle condition for oriented percolation in high dimensions
cites work: The contact process on trees
cites work: Critical behavior of percolation and Markov fields on branching planes
cites work: The contact process on a tree: Behavior near the first phase transition
cites work: A note on inhomogeneous percolation
DOI: 10.1214/AOP/1023481113
description: scientific article; zbMATH DE number 1130488