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

Mikhail Basok

TL;DR

The paper extends Kenyon's height-function framework for the dimer model from simply and doubly connected domains to arbitrary bordered Riemann surfaces. It shows that Kenyon's height-identity correlations can be realized as expectations in a compactified Gaussian free field, with a discrete component encoded by twists in the Jacobian via flat and spin line bundles on Schottky doubles. The core method combines a variation identity for observables, a square-root gauge trick, and a structural gauge equivalence that identifies the relevant correlation kernels with Cauchy kernels of perturbed Cauchy–Riemann operators. The results provide a universal description of height fluctuations on complex topologies, including explicit computations of boundary height gaps and shifts for several geometries, and connect dimer scaling limits to compactified free fields. Overall, the work solidifies the link between discrete complex analysis, Riemann-surface techniques, and the probabilistic structure of the dimer height field on generic bordered surfaces.

Abstract

In a 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. Scaling limits of these identities (if exist) 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, under natural assumptions, these correlations always define a Gaussian free field in a simply connected domain. This was generalized to doubly connected domains in the recent work [Chelkak, Deiman, 25], where the field is shown to be a sum of Gaussian free field and a discrete Gaussian component. We generalize this result further to arbitrary bordered Riemann surfaces.

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

TL;DR

The paper extends Kenyon's height-function framework for the dimer model from simply and doubly connected domains to arbitrary bordered Riemann surfaces. It shows that Kenyon's height-identity correlations can be realized as expectations in a compactified Gaussian free field, with a discrete component encoded by twists in the Jacobian via flat and spin line bundles on Schottky doubles. The core method combines a variation identity for observables, a square-root gauge trick, and a structural gauge equivalence that identifies the relevant correlation kernels with Cauchy kernels of perturbed Cauchy–Riemann operators. The results provide a universal description of height fluctuations on complex topologies, including explicit computations of boundary height gaps and shifts for several geometries, and connect dimer scaling limits to compactified free fields. Overall, the work solidifies the link between discrete complex analysis, Riemann-surface techniques, and the probabilistic structure of the dimer height field on generic bordered surfaces.

Abstract

In a 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. Scaling limits of these identities (if exist) 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, under natural assumptions, these correlations always define a Gaussian free field in a simply connected domain. This was generalized to doubly connected domains in the recent work [Chelkak, Deiman, 25], where the field is shown to be a sum of Gaussian free field and a discrete Gaussian component. We generalize this result further to arbitrary bordered Riemann surfaces.

Paper Structure

This paper contains 27 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Main result: multiply connected domains
  3. Main result: bordered Riemann surfaces
  4. Discussion of the proof
  5. Organization of the paper
  6. Preliminaries on Riemann surfaces
  7. Harmonic differentials and Hodge decomposition
  8. Simplicial basis in $H_1(\Sigma,{\mathbb{Z}})$ and normalized differentials
  9. Line bundles on Riemann surfaces
  10. Sections of a line bundle
  11. Metric and curvature. Riemann--Roch theorem
  12. Flat line bundles and Cauchy--Riemann operators
  13. Jacobian of a Riemann surface
  14. Cauchy kernel of a line bundle
  15. Spin line bundles
  16. ...and 12 more sections

Key Result

Theorem A

Assume that the functions $f^{[\pm,\pm]}$ defining $U_m$ are multivalued functions on $\Omega\times\Omega\smallsetminus\mathrm{diag}$ satisfying the following properties: Then there exist $c_0\in {\mathbb{R}}^n$ and $\mathfrak{m}\in {\mathbb{Z}}^n$ such that for the compactified free field with the shift $c_0$ we have $\mathop{\mathrm{\mathbb{E}}}\nolimits[\exp(\pi i\mathfrak{m}\cdot (c-c_0))]\ne

Theorems & Definitions (37)

  • Theorem A
  • Definition 1.1
  • Theorem B
  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 3.1
  • ...and 27 more