- 99-160 Olle Haggstrom, Roberto H. Schonmann and Jeffrey E. Steif
- The Ising Model on Diluted Graphs and Strong Amenability
(295K, postcript)
May 10, 99
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Abstract. 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 $p$.
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