Kenyon's identities for the height function and compactified free field in the dimer model

Abstract

In his seminal paper published in 2000 Kenyon developed a method to study the height function of the planar dimer model via discrete complex analysis tools. The core of this method is a set of identities representing height correlations through the inverse Kasteleyn operator. In a general setup, such as considered in [Chelkak, Laslier, Russkikh, 23, 22], scaling limits of these identities produce a set of correlation functions written in terms of a Dirac Green's kernel with unknown boundary conditions. It was proven in [Chelkak, Laslier, Russkikh, 23] that, in a simply connected domain, these correlation functions always coincide with correlation functions of the Gaussian free field given that they satisfy some natural a priori assumptions. This was generalized to doubly connected domains in the recent work [Chelkak, Deiman, 26], where correlations are shown to be the correlations of a sum of Gaussian free field and a discrete Gaussian component. We generalize this result further to arbitrary bordered Riemann surfaces.