Rigidity of the Critical Phases on a Cayley Tree

We discuss statistical mechanics on non-amenable graphs, and we study the features of the phase transition, which are due to non-amenability. For the Ising model on the usual lattice it is known that uctuations of magnetization are much less likely in the states with non-zero magnetic eld than in the pure states with zero eld. We show that on the Cayley tree the corresponding uctuations have the same order. Key words and phrases: tree, non-amenable graph, Ising model, large deviations, droplet. 1 Introduction and statement of results This paper is a result of our attempt to understand the nature of the statistical mechanics on non-amenable graphs, and in particular the specic nature of the phenomenon of the rst order phase transition, whose specics is due to nonamenability. The topic of the statistical mechanics on non-amenable graphs is a modern growing eld, and for its present status the reader can consult, e.g., the papers [Ly] and [S]. Let G be an innite, locally nite,...

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