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

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 beta is obtained as the unique solution of an algebraic equation in the variables (tanh(beta J_e)_{ein E(G)}. This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices