1209.0951 (David Cimasoni et al.)

The critical temperature for the Ising model on planar doubly periodic
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David Cimasoni, Hugo Duminil-Copin

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.

View original: http://arxiv.org/abs/1209.0951