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 for 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 $p$.
"The Ising model on diluted graphs and strong amenability." Ann. Probab. 28 (3) 1111 - 1137, July 2000. https://doi.org/10.1214/aop/1019160327