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

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

2008-12-19

arxiv.org/abs/0812.3848v1

Mathematics

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35 pages, 8 figures

Scientific paper

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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