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

TL;DR

This paper extends Kenyon's identities for the height function in the dimer model to arbitrary bordered Riemann surfaces, connecting discrete models with continuous Gaussian free fields in complex analysis.

Contribution

It generalizes the correlation identities and boundary condition results from simply and doubly connected domains to all bordered Riemann surfaces.

Findings

Correlation functions match Gaussian free field predictions

Boundary conditions are characterized for general Riemann surfaces

Extension of Kenyon's identities to complex topologies

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.