Amenability and phase transition in the Ising model
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~\({\mathbb Z}^d\).
scientific article; zbMATH DE number 1309020