Translation-Invariant Gibbs States of Ising model: General Setting

TL;DR

This paper proves that on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of plus and minus states, extending known results to more general graphs and interactions.

Contribution

It establishes the convex decomposition of Gibbs states for the Ising model on general graphs with automorphism-invariant interactions, using the random current representation.

Findings

Gibbs states are convex combinations of plus and minus states.

Continuity of magnetization at non-critical temperatures.

Differentiability of the free energy.

Abstract

We prove that at any inverse temperature $\beta$ and on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of the plus and minus states. This is obtained for a general class of interactions, that is automorphism-invariant and irreducible coupling constants. The proof uses the random current representation of the Ising model. The result is novel when the graph is not $\mathbb{Z}^d$, or when the graph is $\mathbb{Z}^d$ but endowed with infinite-range interactions, or even $\mathbb{Z}^2$ with finite-range interactions. Among the corollaries of this result, we can list continuity of the magnetization at any non-critical temperature, the differentiability of the free energy, and the uniqueness of FK-Ising infinite-volume measures.