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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 βfor 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(βJ_e))_{e\in E(G)}. This is achieved by studying the high-temperature expansion of the model using Kac-Ward matrices.

preprint2012arXivOpen access
David CimasoniHugo Duminil-Copin
math-phmath.PRmath.MP
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David CimasoniHugo Duminil-Copin

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