The Z-invariant Ising model via dimers
Résumé
The $Z$-invariant Ising model [3] is defined on an isoradial graph and has coupling constants depending on an elliptic parameter $k$. When $k=0$ the model is critical, and as $k$ varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers [7, 8] to the full $Z$-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of [8]: it involves a local function and the massive discrete exponential function introduced in [10]. This shows in particular that $Z$-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the $Z$-invariant spanning forests of [10], and deduce that the two models have the same order two phase transition in $k$. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this $Z$-invariant Ising model.
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