The Ising model on diluted graphs and strong amenability

• 成果类型: Article
• 署名作者: Häggström, O; Schonmann, RH; Steif, JE
• 署名单位: Chalmers University of Technology; University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 1111-1137
• 关键词: random spin systems; infinite clusters; random-walks; invariant percolation; branching planes; PHASE-TRANSITION; markov-fields; bethe lattice; trees; MONOTONICITY

摘要

Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field. we show that for nonamenable graphs, for Bernoulli percolation with p close to 1, all the infinite clusters have persistent transition. On the other hand, we show that for transitive amenable graphs, the infinite clusters fbr any stationary percolation do not have persistent transition This extends a result of Georgii for the cubic lattice. A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (i.e., their anchored Cheeger constant is 0). Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter p, on an arbitrary bounded degree graph, is a continuous function of p.