The critical Z-invariant Ising model via dimers: the periodic case
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 , triangular and honeycomb lattice at the critical temperature. Fisher (J Math Phys 7:1776–1781, 1966) introduced a correspondence between...
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Veröffentlicht in: | Probability theory and related fields 2010-07, Vol.147 (3-4), p.379-413 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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
, 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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ISSN: | 0178-8051 1432-2064 |
DOI: | 10.1007/s00440-009-0210-1 |