On the scaling of the chemical distance in long-range percolation models
type: Item
instance of: scholarly article
title: On the scaling of the chemical distance in long-range percolation models
zbMATH Open document ID: 1072.60084
DOI: 10.1214/009117904000000577
author: M. Biskup
published in: The Annals of Probability
publication date: 2005-03-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0304418
review text: The paper deals with long-range percolation on \({\mathbb Z}^d\) where two sites \(x\) and \(y\) get connected with probability \(p_{xy}=| x-y| ^{-s+o(1)}\) as \(| x-y| \rightarrow \infty\). The interest focuses on the scaling of the graph distance (or chemical distance) between two remote sites. The introduction gives an overview of previous works concerning the five distinct regimes marked by the position of \(s\) relative to \(d\) and \(2d\) and discusses the relationship with the so-called ``small-world'' phenomena. The main result of the paper is a proof of a polylogarithmic estimation on the asymptotic behavior of the graph distance when \(d<s<2d\) and the random graph contains a unique infinite component. The study of percolation in finite boxes is an essential tool for the proof.
reviewed by: Dominique Lépingle
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 82B28
zbMATH DE Number: 2148681
zbMATH Keywords: long-range percolation
zbMATH Keywords: chemical distance
zbMATH Keywords: renormalization
zbMATH Keywords: small-world phenomena
MaRDI profile type: Publication
OpenAlex ID: W2065630310
arXiv ID: math/0304418
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description: scientific article