Continuity of the Ising Phase Transition on Nonamenable Groups

Abstract

We prove rigorously that the ferromagnetic Ising model on any nonamenable Cayley graph undergoes a continuous (second-order) phase transition in the sense that there is a unique Gibbs measure at the critical temperature. The proof of this theorem is quantitative and also yields power-law bounds on the magnetization at and near criticality. Indeed, we prove more generally that the magnetization σ o + β,h is a locally Hölder-continuous function of the inverse temperature β and external field h throughout the non-negative quadrant (β, h) ∈ [0, ∞) 2 . As a second application of the methods we develop, we also prove that the free energy of Bernoulli percolation is twice differentiable at p c on any transitive nonamenable graph.