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

   
@article{EJP2352,
	author = {David Cimasoni and Hugo Duminil-Copin},
	title = {The critical temperature for the Ising model on planar doubly periodic graphs},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {18},
	year = {2013},
	keywords = {Ising model;critical temperature;weighted periodic graph;Kac-Ward matrices;Harnack curves},
	abstract = {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.},
	pages = {no. 44, 1-18},
	issn = {1083-6489},
	doi = {10.1214/EJP.v18-2352},    
        url = {http://ejp.ejpecp.org/article/view/2352}}