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.