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We provide a simple characterization of the critical temperature for theIsing model on an arbitrary planar doubly periodic weighted graph. Moreprecisely, the critical inverse temperature \beta for a graph G with couplingconstants (J_e)_{e\in E(G)} is obtained as the unique solution of a linearequation in the variables (\tanh(\beta J_e))_{e\in E(G)}. This is achieved bystudying the high-temperature expansion of the model using Kac-Ward matrices.

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