Continuity of the Ising phase transition on nonamenable groups

TL;DR

This paper proves that the ferromagnetic Ising model on nonamenable Cayley graphs exhibits a continuous phase transition with unique Gibbs measure at criticality, providing power-law bounds and regularity results.

Contribution

It establishes the continuity of the phase transition and regularity of magnetization and free energy on nonamenable graphs, with new quantitative bounds and methods.

Findings

Unique Gibbs measure at critical temperature

Power-law bounds on magnetization near criticality

Free energy is twice differentiable at critical point

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 $\langle \sigma_o \rangle_{\beta,h}^+$ is a locally H\"older-continuous function of the inverse temperature $\beta$ and external field $h$ throughout the non-negative quadrant $(\beta,h)\in [0,\infty)^2$. 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.