Critical values in a long-range percolation on spaces like fractals
type: Item
instance of: scholarly article
title: Critical values in a long-range percolation on spaces like fractals
zbMATH Open document ID: 1114.82015
author: Jun Misumi
published in: Journal of Statistical Physics
publication date: 2007-02-16T00:00:00Z
review text: The paper considers a long-range percolation process on a general countable discrete space \(X\) where the distinct points \(x\), \(y\) are connected with probability \(p(x,y)\) satisfying \[ \lim_{\rho(x,y)\to\infty} {p(x,y)\over\beta \rho(x,y)^{-\alpha}}=1, \] and \(\rho(x,y)\) is a symmetric function playing the role of the distance. It is also assumed that the space \(X\) together with the function \(\rho\) satisfies a type of volume growth condition, which is used to define ``dimension'' \(D\) of the space \(X\). Under some additional assumptions on the structure of \((X,\rho)\), the author proves that 1. If \(\alpha < D\), then all points in \(X\) are connected with probability 1. 2. If \(\alpha > D\), then there exists \(p\) such that there is no infinite percolation cluster. 3. If \(\alpha > 2D\), then there is no infinite percolation cluster. This parallels results obtained for the long-range percolation on \(\mathbb Z\) [e.g. \textit{M. Aizenman} and \textit{C. M. Newman}, Commun. Math. Phys. 107, 611--647 (1986; Zbl 0613.60097) or \textit{C. M. Newman} and \textit{L. S. Schulman}, Commun. Math. Phys. 104, 547--571 (1986; Zbl 0604.60097)]. The results are then applied to the long-range percolation on some fractals.
reviewed by: Jiří Černý
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B26
zbMATH DE Number: 5127775
zbMATH Keywords: Long-range percolation
zbMATH Keywords: phase transition
zbMATH Keywords: fractals
MaRDI profile type: Publication
full work available at URL: s10955-006-9243-3
OpenAlex ID: W2027427977
cites work: Discontinuity of the percolation density in one dimensional \(1/| x- y| ^ 2\) percolation models
cites work: On the chemical distance for supercritical Bernoulli percolation
cites work: Transience, recurrence and critical behavior for long-range percolation
cites work: The diameter of long-range percolation clusters on finite cycles
cites work: Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12,\dots
cites work: On the scaling of the chemical distance in long-range percolation models
cites work: The diameter of a long-range percolation graph
cites work: On the connectedness of a random graph
cites work: When are random graphs connected
cites work: One dimensional \(1/| j-i| ^ s\) percolation models: The existence of a transition for s\(\leq 2\)
cites work: Existence of phase transition of percolation on Sierpiński carpet lattices
cites work: Non-existence of phase transition of oriented percolation on Sierpiński carpet lattices
DOI: 10.1007/S10955-006-9243-3
description: scientific article; zbMATH DE number 5127775