Translation-invariant Gibbs states of the Ising model: General setting

We prove that at any inverse temperature . 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. The theorem is equivalent with the differentiability of the free energy with respect to the temperature at any temperature. 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 Zd, or when the graph is Zd but endowed with infinite-range interactions, or even Z2 with finite-range interactions. Among the other corollaries of this result, we can list continuity of the magnetization at any noncritical temperature and the uniqueness of FK-Ising infinite-volume measures at any temperature.