The critical Z-invariant Ising model via dimers: the periodic case
  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 square, triangular and honeycomb lattice at the critical temperature. Fisher 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. 
C\'edric Boutillier, B\'eatrice de Tili\`ere