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Journal Articles Probability Theory and Related Fields Year : 2010

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.
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Dates and versions

hal-00354294 , version 1 (04-01-2022)

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Cédric Boutillier, Béatrice de Tilière. The critical Z-invariant Ising model via dimers: the periodic case. Probability Theory and Related Fields, 2010, 147 (3-4), pp.379-413. ⟨10.1007/s00440-009-0210-1⟩. ⟨hal-00354294⟩
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