Translation-Invariant Gibbs States of Ising model: General Setting

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.