The critical temperature for the Ising model on planar doubly periodic graphs

Abstract

We provide a simple characterization of the critical temperature for the Ising model on an arbitrary planar doubly periodic weighted graph. More precisely, the critical inverse temperature β for a graph G with coupling constants (Je)e∈E(G) is obtained as the unique solution of an algebraic equation in the variables (tanh(βJe))e∈E(G). This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices.