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

• 成果类型: Article
• 署名作者: Boutillier, Cedric; de Tiliere, Beatrice
• 署名单位: Universite Paris Cite; Sorbonne Universite; University of Neuchatel
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-009-0210-1
• 发表日期: 2010
• 页码: 379-413
• 关键词: STATISTICS; lattice; graphs

摘要

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 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.