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 $p$.