Continuity of the Ising Phase Transition on Nonamenable Groups.

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. [ABSTRACT FROM AUTHOR]