We introduce a one-parameter family of massive Laplacian operators (∆ m(k) ) k∈[0,1) defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of ∆ m(k) , the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at k = 0, thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon [Ken02] are critical. We prove that the massive Laplacian operators (∆ m(k) ) k∈(0,1) provide a one-parameter family of Z-invariant rooted spanning forest models. When the isoradial graph is moreover Z 2 -periodic, we consider the spectral curve of the characteristic polynomial of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus 1. We further show that every Harnack curve of genus 1 with (z, w) ↔ (z −1 , w −1 ) symmetry arises from such a massive Laplacian.

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. Mathematics Subject Classification (2000) 82B20 IntroductionIn [11], Fisher introduced a correspondence between the two-dimensional Ising model defined on a graph G, and the dimer model defined on a decorated version of this graph. Since then, dimer techniques have been a powerful tool for solving pertinent questions about the Ising model, see for example the paper of Kasteleyn [14], and the book of Mc Coy and Wu [23]. In this paper, we follow this approach to the Ising model.where K (z, w) is the Fourier transform of the infinite Kasteleyn matrix K (it is a matrix of size V (G 1 ) × V (G 1 )). 2. The weak limit of the Boltzmann measures P n defines a translation invariant ergodic Gibbs measure P. The probability of occurrence of a subset of edges e 1 = u 1 v 1 , . . . , e m = u m v m of G, in a dimer configuration of G chosen with respect to the Gibbs measure P, is: P(e 1 , . . . , e m ) = m i=1 K u i ,v i Pf (K −1 ) t e 1 ,...,e m , P J (σ ) = 1 Z J exp e=uv∈E(G) J e σ u σ v , where Z J = σ ∈{−1,1} V (G) exp e=uv∈E(G)

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 Zinvariant 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].