Electronic Journal of Probability

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

David Cimasoni and Hugo Duminil-Copin

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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 an algebraic 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.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 44, 18 pp.

Accepted: 28 March 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Ising model critical temperature weighted periodic graph Kac-Ward matrices Harnack curves

This work is licensed under a Creative Commons Attribution 3.0 License.


Cimasoni, David; Duminil-Copin, Hugo. The critical temperature for the Ising model on planar doubly periodic graphs. Electron. J. Probab. 18 (2013), paper no. 44, 18 pp. doi:10.1214/EJP.v18-2352. https://projecteuclid.org/euclid.ejp/1465064269

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