Progress in high-dimensional percolation and random graphs
type: Item
instance of: scholarly article
title: Progress in high-dimensional percolation and random graphs
zbMATH Open document ID: 1445.60003
DOI: 10.1007/978-3-319-62473-0
author: Markus Heydenreich
author: Remco van der Hofstad
published in: CRM Short Courses
publication date: 2017-08-02T00:00:00Z
review text: This is a textbook on percolation. The authors consider a lattice -- let us focus on \(\mathbb{Z}^{d}\) here -- where each vertex is adjacent to each of the \(2d\) vertices which differ from it in one coordinate only and then by \(\pm 1\). Percolation is basically the situation where each bond (i.e., edge) is occupied with probability \(p\) and vacant with probability \(1-p\) independent of all other bonds and we study the behaviour of the occupied bonds. For example, for what values of \(p\) is there an infinite cluster of occupied bonds? Letting \(\mathcal{C}(x)\) denote the cluster of a vertex \(x\), we let \(\theta(p)=\mathbb{P}_{p}\left(\vert\mathcal{C}(x)\vert=\infty\right)\). This is clearly independent of \(x\) by transitivity but dependent on \(p\). We let \(p_{c}=\inf \{p: \theta(p)>0\}\) be the critical probability where there is a change from finite clusters to (at least) one infinite cluster. Much of the interest is in values of \(p\) close to \(p_{c}\). For example, a famous question, unsolved in general, is to show that, when \(d\geq 2\), there is no infinite cluster when \(p=p_{c}\). One of the main new contributions of the book under review is to give a self-contained proof that for large enough \(d\) this statement is true (the corresponding result for \(d=2\) is due to \textit{H. Kesten} [Probab. Theory Relat. Fields 73, 369--394 (1986; Zbl 0584.60098)]). It is expected from physical considerations that \(\theta(p)\sim (p-p_{c})^{\beta}\) for some constant \(\beta\) as \(p\downarrow p_{c}\). We might also be interested in \(\chi(p)\), the expected size of the cluster of the origin. Again, physics predicts (but does not prove) the existence of a constant \(\gamma\) such that \(\chi(p)\sim (p-p_{c})^{-\gamma}\). We could further consider \(\chi^{f}(p)\), the size of the cluster of the origin conditional on its being finite, where again it is predicted that \(\chi^{f}(p)\sim (p-p_{c})^{-\delta}\) . (Soma caution in needed about the multiple interpret…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 05C80
zbMATH DE Number: 6755664
zbMATH Keywords: percolation
zbMATH Keywords: mean field behaviour
zbMATH Keywords: critical exponent
zbMATH Keywords: high dimensions
zbMATH Keywords: triangle condition
zbMATH Keywords: infrared bound
zbMATH Keywords: lace expansion
reviewed by: David B. Penman
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-62473-0
OpenAlex ID: W2769174420
description: scientific article; zbMATH DE number 6755664