The critical Z-invariant Ising model via dimers: the periodic case

Abstract : We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical square, triangular and honeycomb lattice at the critical temperature. Fisher introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model. We prove that the dimer characteristic polynomial is equal (up to a constant) to the critical Laplacian characteristic polynomial, and defines a Harnack curve of genus 0. We prove an explicit expression for the free energy, and for the Gibbs measure obtained as weak limit of Boltzmann measures.
Type de document :
Pré-publication, Document de travail
35 pages, 8 figures. 2008
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Contributeur : Cédric Boutillier <>
Soumis le : lundi 19 janvier 2009 - 14:53:53
Dernière modification le : mercredi 21 mars 2018 - 18:56:48

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  • HAL Id : hal-00354294, version 1
  • ARXIV : 0812.3848



Cédric Boutillier, Béatrice De Tilière. The critical Z-invariant Ising model via dimers: the periodic case. 35 pages, 8 figures. 2008. 〈hal-00354294〉



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