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

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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 \({\mathbb{Z}^2}\) , triangular and honeycomb lattice at the critical temperature. Fisher (J Math Phys 7:1776–1781, 1966) 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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Correspondence to Cédric Boutillier.

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Boutillier, C., de Tilière, B. The critical Z-invariant Ising model via dimers: the periodic case. Probab. Theory Relat. Fields 147, 379–413 (2010). https://doi.org/10.1007/s00440-009-0210-1

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Mathematics Subject Classification (2000)

  • 82B20