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 for a graph G with coupling constants (J_e)_{e\in E(G)} is obtained as the unique solution of a linear equation in the variables (\tanh(\beta J_e))_{e\in E(G)}. This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices. 
David Cimasoni, Hugo Duminil-Copin
math-ph math.MP math.PR