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

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.

💡 Research Summary

The paper revisits and substantially extends the celebrated identities introduced by Richard Kenyon in his 2000 work, which relate height‑function correlations in the planar dimer model to entries of the inverse Kasteleyn matrix. Kenyon’s original identities (often written as a determinant of a matrix built from the inverse Kasteleyn operator) provide a bridge between discrete combinatorics and continuous complex analysis: after a suitable scaling limit the inverse Kasteleyn matrix is expected to converge to the Green’s kernel of a Dirac operator on the underlying domain. In the simply‑connected Temperleyan setting this leads to the Gaussian free field (GFF) with Dirichlet boundary conditions, a result proved by Kenyon and later refined by many authors.

Recent works by Chelkak, Laslier, and Russkikh (CLR23, CLR22) introduced the language of t‑embeddings (perfect t‑embeddings) and showed that, after a Coulomb gauge fixing, the rescaled inverse Kasteleyn matrix converges to a Dirac Green’s kernel defined on a Lorentz‑minimal surface in ℝ²,². The kernel’s boundary conditions, however, remain a priori unknown and can be highly non‑trivial when the domain is multiply connected or when the Temperleyan boundary conditions are altered (e.g., different “colors” on distinct boundary components). In such cases the kernel is not conformally covariant, and the resulting height‑field cannot be identified directly with a GFF.

The present work tackles precisely this difficulty and pushes the analysis to its most general topological setting: arbitrary bordered Riemann surfaces, possibly with handles, multiple boundary components, and conical singularities in the underlying flat metric. The author’s strategy consists of three main ingredients:

1. Generalized Kenyon Identities.
   Starting from the determinant representation of height‑difference moments (Equation 1.1), the author rewrites the scaling limit as a sum over spin configurations of determinants built from the limiting Dirac kernel components (f^{\pm,\pm}). The resulting expression (Equation 1.2) involves a product of holomorphic/anti‑holomorphic differentials (dz) or (d\bar z). This formulation is robust enough to survive gauge transformations and to be defined on any Riemann surface once a suitable line bundle (the spin bundle) is chosen.

2. Compactified Free Field Decomposition.
   For a multiply‑connected planar domain (\Omega) with boundary components (B_0,\dots,B_n), the height field is shown to decompose as
   \