Amenability and phase transition in the Ising model
type: Item
instance of: scholarly article
title: Amenability and phase transition in the Ising model
zbMATH Open document ID: 0940.60093
DOI: 10.1023/A:1021690414168
author: Johan Jonasson
author: Jeffrey E. Steif
published in: Journal of Theoretical Probability
publication date: 2000-07-12T00:00:00Z
review text: The authors consider the Ising model with external field \(h\) and coupling constant \(J\) on an infinite connected graph \(G\) of uniformly bounded degree and relate phase transitions in the Ising model to amenability properties of \(G\). More precisely, they show, using a variant of Peierls argument, that if the graph \(G\) is nonamenable, then the Ising model on \(G\) exhibits phase transition for some \(J\in[0,\infty)\) and \(h\neq 0\) (an infinite graph \(G=(V,E)\) is called nonamenable, if for any finite \(W\subset{V}\) the boundary \(\partial W\) is of the same order as \(W\) itself; i.e., when the ratio of cardinalities \(|\partial W|/|W|\) is uniformly positive for all such \(W\)). This generalises a similar result known before for the regular Cayley trees [see, e.g., \textit{H.-O. Georgii}, ``Gibbs measures and phase transitions'' (1988; Zbl 0657.60122)]. On the other hand, if \(G\) is amenable and quasi-transitive, then in the Ising model on \(G\) no phase transitions occur for any \(h>0\) and \(J\in[0,\infty)\) (an infinite graph \(G=(V,E)\) is called quasi-transitive, if the automorphism group of \(G\) acting on the set \(V\) of vertices has a finite number of orbits). The proof is based on a variant of the classical convexity and differentiability of a pressure argument for the \(d\)-dimensional lattice~\({\mathbb Z}^d\).
reviewed by: Ostap Hryniv
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B26
zbMATH DE Number: 1309020
zbMATH Keywords: Ising model
zbMATH Keywords: Cayley graphs
zbMATH Keywords: amenability
zbMATH Keywords: Markov operators
zbMATH Keywords: phase transition
MaRDI profile type: Publication
full work available at URL: a:1021690414168
OpenAlex ID: W1496074149
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description: scientific article; zbMATH DE number 1309020