Dimers and families of Cauchy-Riemann operators I

Abstract

In the dimer model, a configuration consists of a perfect matching of a fixed graph. If the underlying graph is planar and bipartite, such a configuration is associated to a height function. For appropriate “critical” (weighted) graphs, this height function is known to converge in the fine mesh limit to a Gaussian free field, following in particular Kenyon’s work. In the present article, we study the asymptotics of smoothed and local field observables from the point of view of families of Cauchy-Riemann operators and their determinants. This allows one in particular to obtain a functional invariance principle for the field; characterise completely the limiting field on toroidal graphs as a compactified free field; analyze electric correlators; and settle the Fisher-Stephenson conjecture on monomer correlators. The analysis is based on comparing the variation of determinants of families of (continuous) Cauchy-Riemann operators with that of their discrete (finite dimensional) approximations. This relies in turn on estimating precisely inverting kernels, in particular near singularities. In order to treat correlators of “singular” local operators, elements of (multiplicatively) multivalued discrete holomorphic functions are discussed.