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

TL;DR

This paper provides an explicit local formula for the inverse Kasteleyn matrix of the dimer model associated with critical Z-invariant Ising models, enabling a new proof of Baxter's free energy formula using local geometric data.

Contribution

It introduces a local formula for the inverse Kasteleyn matrix of the dimer model corresponding to critical Z-invariant Ising models, connecting geometry with statistical mechanics.

Findings

Explicit local formula for inverse Kasteleyn matrix

New proof of Baxter's free energy formula

Connection between dimer models and Ising model geometry

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].