# The critical Z-invariant Ising model via dimers: locality property

Abstract : We study a large class of critical two-dimensional Ising models, namely critical Z-invariant Ising models. Fisher [Fis66] 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, consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph. Our main result is an explicit local formula for the inverse Kasteleyn matrix, in the spirit of [Ken02], as a contour integral of the discrete exponential function of [Mer01a,Ken02] multiplied by a local function. Using results of [BdT08] and techniques of [dT07b,Ken02], this yields an explicit local formula for a natural Gibbs measure, and a local formula for the free energy. As a corollary, we recover Baxter's formula for the free energy of the critical Z-invariant Ising model [Bax89], and thus a new proof of it. The latter is equal, up to a constant, to the logarithm of the normalized determinant of the Laplacian obtained in [Ken02].
Type de document :
Pré-publication, Document de travail
55 pages, 29 figures. 2009
Domaine :

https://hal.archives-ouvertes.fr/hal-00361365
Contributeur : Cédric Boutillier <>
Soumis le : vendredi 13 février 2009 - 19:20:40
Dernière modification le : mardi 30 mai 2017 - 01:07:54

### Identifiants

• HAL Id : hal-00361365, version 1
• ARXIV : 0902.1882

### Citation

Cédric Boutillier, Béatrice De Tilière. The critical Z-invariant Ising model via dimers: locality property. 55 pages, 29 figures. 2009. 〈hal-00361365〉

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