The Ising model on diluted graphs and strong amenability

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 pre sence of a nonzero external field. we show that for nonamenable graphs, for Bernoulli percolation with p close to 1, all the infinite clusters have pe rsistent transition. On the other hand, we show that for transitive amenabl e graphs, the infinite clusters fbr any stationary percolation do not have persistent transition This extends a result of Georgii for the cubic lattic e. A geometric consequence of this latter fact is that the infinite cluster s are strongly amenable (i.e., their anchored Cheeger constant is 0). Final ly we show that the critical temperature for the Ising model with no extern al field on the infinite clusters of Bernoulli percolation with parameter p , on an arbitrary bounded degree graph, is a continuous function of p.