2009/02/11 by Cédric Boutillier, Béatrice de Tilière
Mathematics · Physics and Astronomy · #Exponential function #Generating function #Gibbs measure #Ising model #Laplacian matrix #Logarithm #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Square-lattice Ising model #Statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:82B20
paper · pdf · doi:10.1007/s00220-010-1151-3
published as Comm. Math. Phys. 301 (2011), no.2, 473-516 · 55 pages, 29 figures
arxiv created 2009/02/11 · openalex publication_date 2010/11/09 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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].