Ising models on hyperbolic graphs
type: Item
instance of: scholarly article
title: Ising models on hyperbolic graphs
zbMATH Open document ID: 0926.60079
DOI: 10.1007/BF02175564
author: C. Chris Wu
published in: Journal of Statistical Physics
publication date: 1999-11-18T00:00:00Z
review text: The author studies the nearest-neighbour ferromagnetic Ising model on a hyperbolic graph \(G\) that can be thought as a discretisation of the hyperbolic plane. More formally, the graph \(G\) is constructed from a rooted \(k\)-tree \(T_k\) by adding edges connecting some equal-level sites in such a way that at the level \(l\geq 1\) exactly \(k^l-1\) edges are added (as a result, for \(l\geq 1\) there are \(2\) vertices of index \(k+2\) and \(k^l-2\) vertices of index \(k+3\)). It is shown that the Ising model on \(G\) exhibits multiple phase transitions. Namely, let \(\nu^+\), \(\nu^-\), and \(\nu^f\) denote the Gibbs states with plus, minus, and free boundary conditions, respectively, and let \(M\) be the mean of the spin at the origin w.r.t.~\(\nu^+\). Then, for \(k\geq 8\), there exist \(\beta_c\), \(\beta_c'\), and \(\beta_c''\) satisfying \(0<\beta_c<\beta_c'\leq\beta_c''\) such that the following holds: 1) for \(\beta<\beta_c\), \(M=0\) and there is only one Gibbs state; 2) for \(\beta_c<\beta<\beta_c'\), \(M>0\) and \(\nu^f\neq(\nu^++\nu^-)/2\), i.e. there are more than two extremal Gibbs states; 3) for \(\beta>\beta_c''\), \(M>0\) and \(\nu^f=(\nu^++\nu^-)/2\). The proof is based on the Fortuin-Kasteleyn (FK) representation of Ising models and the comparison inequalities between FK and independent percolation measures. It is conjectured that \(\beta_c'=\beta_c''\) and that all the results are valid for \(k\geq 2\).
reviewed by: Ostap Hryniv
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B20
Mathematics Subject Classification ID: 82B26
zbMATH DE Number: 1279710
zbMATH Keywords: Ising model
zbMATH Keywords: hyperbolic graph
zbMATH Keywords: multiple phase transitions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02175564
OpenAlex ID: W1969566182
description: scientific article; zbMATH DE number 1279710