Abstract
We consider the Ising model with external field h and coupling constant J on an infinite connected graph G with uniformly bounded degree. We prove that if G is nonamenable, then the Ising model exhibits phase transition for some h≠0 and some J<∞. On the other hand, if G is amenable and quasi-transitive, then phase transition cannot occur for h≠0. In particular, a group is nonamenable if and only if the Ising model on one (all) of its Cayley graphs exhibits a phase transition for some h≠0 and some J<∞.
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Jonasson, J., Steif, J.E. Amenability and Phase Transition in the Ising Model. Journal of Theoretical Probability 12, 549–559 (1999). https://doi.org/10.1023/A:1021690414168
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DOI: https://doi.org/10.1023/A:1021690414168