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Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models

Tomas Berggren, Matthew Nicoletti

TL;DR

This work analyzes height fluctuations in Aztec diamond dimer models with doubly periodic edge weights, showing that the centered height field decomposes into a Gaussian free field on the multiply connected liquid region and a random harmonic component whose boundary data form an $N$-dependent discrete Gaussian. The authors develop a two-pronged approach: an analytic steepest-descent analysis of exact inverse-Kasteleyn formulas on the spectral curve to capture bulk, edge, and facet contributions, and an algebraic step using theta functions and Fay's identity to identify the discrete Gaussian structure via cumulants. They prove moment convergence to a GFF on the uniformized domain and show the discrete components are asymptotically independent from the GFF, with joint cumulants described by derivatives of theta functions and an $N$-dependent shift in the discrete Gaussian distribution. The results extend the universality of discrete Gaussian fluctuations to higher-genus, multiply connected liquid regions induced by gaseous facets, and they connect refined partition-function data to the limit shape through the spectral-curve formalism. The findings provide a rigorous, theta-function–driven description of height fluctuations in a broad class of periodic dimer models with topologically nontrivial liquid regions.

Abstract

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected liquid region and a harmonic function with random liquid-gas boundary values. The boundary values are jointly distributed as a discrete Gaussian random vector. This discrete Gaussian distribution maintains a quasi-periodic dependence on $N$, a phenomenon also observed in multi-cut random matrix models.

Gaussian Free Field and Discrete Gaussians in Periodic Dimer Models

TL;DR

This work analyzes height fluctuations in Aztec diamond dimer models with doubly periodic edge weights, showing that the centered height field decomposes into a Gaussian free field on the multiply connected liquid region and a random harmonic component whose boundary data form an -dependent discrete Gaussian. The authors develop a two-pronged approach: an analytic steepest-descent analysis of exact inverse-Kasteleyn formulas on the spectral curve to capture bulk, edge, and facet contributions, and an algebraic step using theta functions and Fay's identity to identify the discrete Gaussian structure via cumulants. They prove moment convergence to a GFF on the uniformized domain and show the discrete components are asymptotically independent from the GFF, with joint cumulants described by derivatives of theta functions and an -dependent shift in the discrete Gaussian distribution. The results extend the universality of discrete Gaussian fluctuations to higher-genus, multiply connected liquid regions induced by gaseous facets, and they connect refined partition-function data to the limit shape through the spectral-curve formalism. The findings provide a rigorous, theta-function–driven description of height fluctuations in a broad class of periodic dimer models with topologically nontrivial liquid regions.

Abstract

We analyze height fluctuations in Aztec diamond dimer models with nearly arbitrary periodic edge weights. We show that the centered height function approximates the sum of two independent components: a Gaussian free field on the multiply connected liquid region and a harmonic function with random liquid-gas boundary values. The boundary values are jointly distributed as a discrete Gaussian random vector. This discrete Gaussian distribution maintains a quasi-periodic dependence on , a phenomenon also observed in multi-cut random matrix models.

Paper Structure

This paper contains 59 sections, 30 theorems, 237 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Motivation and informal description of results
  3. Statement of results
  4. Convergence to the Gaussian free field
  5. Convergence to the discrete Gaussian distribution
  6. Informal statement of the main result
  7. Proof outline
  8. For the analytic part,
  9. The algebraic step
  10. Plan of the paper
  11. Acknowledgments
  12. Background
  13. Coordinates, Kasteleyn matrix, transition matrices, and height function
  14. The spectral curve and assumptions on edge weights
  15. A and B cycles, theta functions, and prime forms
  16. ...and 44 more sections

Key Result

Theorem 1.1

Let $h_N$ denote the height function of a random dimer configuration of a size $k \ell N$ Aztec diamond with $k \times \ell$ doubly periodic edge weights, and let $\tilde{h}_N$ be defined as in eqn:tildeh_def above. For any positive integer $r \geq 2$, consider faces $\{\mathsf{f}_{j,N}\}_{j=1}^r$ a Moreover, $\tilde{h}_N$ and $(Z_1,\dots, Z_g)$ are asymptotically independent in the sense of momen

Figures (15)

  • Figure 1: A perfect matching of a size $N =4$ Aztec diamond. On the right we also show the reference matching and the associated height function.
  • Figure 2: The difference of two independent height functions sampled from a dimer model with $3 \times 3$ periodic weights, with four gaseous facets. (One of the four facets is difficult to see in the picture.) One observes qualitatively more wild behavior in the liquid region, while gaseous facets are approximately flat, though they exhibit (relatively) sparse defects.
  • Figure 3: Two random samples of domino tilings with doubly periodic weights. The liquid region is connected, but not simply connected.
  • Figure 4: A size $4$ Aztec diamond. Left: Its emedding in $\mathbb{R}^2$. Centers of faces have integer coordinates, which gives coordinates on the set of faces. Right: Our convention for indexing the black and white vertices.
  • Figure 5: A size $4 = k \ell 1$ Aztec diamond with $k \times \ell = 2 \times 2$ periodic weights. The edges with no label have weight $1$. Furthermore it is these edges with a negative sign in the Kasteleyn weighting, and also these edges which are used in the reference matching for computing the height function.
  • ...and 10 more figures

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2: Corollary \ref{['eqn:discrete_gauss_corg']} in the text
  • Definition 2.1: Kasteleyn Matrix
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.5: KOS06
  • Lemma 2.6: Proposition 3.1 of BB23
  • Proposition 2.7: Proposition 5.4 of BB23
  • Remark 2.8
  • Definition 2.9
  • ...and 67 more