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Abstract: We study a large class of critical two-dimensional Ising models, namelycritical Z-invariant Ising models.
Fisher Fis66 introduced a correspondencebetween the Ising model and the dimer model on a decorated graph, thus settingdimer techniques as a powerful tool for understanding the Ising model.
In thispaper, we give a full description of the dimer model corresponding to thecritical Z-invariant Ising model, consisting of explicit expressions which onlydepend on the local geometry of the underlying isoradial graph.
Our main resultis an explicit local formula for the inverse Kasteleyn matrix, in the spirit ofKen02, as a contour integral of the discrete exponential function ofMer01a,Ken02 multiplied by a local function.
Using results of BdT08 andtechniques of dT07b,Ken02, this yields an explicit local formula for anatural 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-invariantIsing model Bax89, and thus a new proof of it.
The latter is equal, up to aconstant, to the logarithm of the normalized determinant of the Laplacianobtained in Ken02.



Author: Cédric Boutillier, Béatrice de Tilière

Source: https://arxiv.org/



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