The Z-Dirac and massive Laplacian operators in the Z-invariant Ising model

Abstract

Consider an elliptic parameter k; we introduce a family of Zu-Dirac operators (K(u))u∈C, relate them to the Z-massive Laplacian of [11], and extend to the full Z-invariant case the results of Kenyon [45] on discrete holomorphic and harmonic functions, which correspond to the case k=0. We prove through combinatorial identities, how and why the Zu-Dirac and Z-massive Laplacian operators appear in the Z-invariant Ising model, considering the case of infinite and finite isoradial graphs. More precisely, consider the dimer model on the Fisher graph GF arising from a Z-invariant Ising model. We express coefficients of the inverse Fisher Kasteleyn operator as a function of the inverse Zu-Dirac operator and also as a function of the Z-massive Green function; in particular this proves a (massive) random walk representation of important observables of the Ising model. We prove that the squared partition function of the Ising model is equal, up to a constant, to the determinant of the Z-massive Laplacian operator with specific boundary conditions, the latter being the partition function of rooted spanning forests. To show these results, we relate the inverse Fisher Kasteleyn operator and that of the dimer model on the bipartite graph GQ arising from the XOR-Ising model, and we prove matrix identities between the Kasteleyn matrix of GQ and the Zu-Dirac operator, that allow to reach inverse matrices as well as determinants.