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Temperleyan Domino Tilings with Holes

Matthew Nicoletti

TL;DR

This work analyzes asymptotic height fluctuations for uniformly random domino tilings on multiply connected Temperleyan domains by combining Kenyon's contour-integral moment formulas with theta-function methods on the doubled domain $R$. It proves that the centered height fluctuation, after removing a random harmonic correction from inner-hole boundary data, converges in moments to a Gaussian free field on $U$, while the hole boundary heights converge to a centered discrete Gaussian with shift $e$ and scale matrix $\tau$, with the two components becoming asymptotically independent. The discrete Gaussian arises from random hole boundary data, and the results confirm predictions for multiply connected tilings and suggest universality of discrete Gaussian contributions in such models. The approach builds a bridge between dimer model fluctuations and compactified free-field behavior via the theta-function framework on the associated Riemann surface, extending the understanding of conformal invariance and universality beyond simply connected domains.

Abstract

We analyze asymptotic height function fluctuations in uniformly random domino tiling models on multiply connected Temperleyan domains. Starting from asymptotic formulas derived by Kenyon [arXiv:math-ph/9910002v1], we show that (1) the difference of the centered height function and a harmonic function with boundary values given by the (random) centered hole heights converges in the sense of moments to a Gaussian free field, which is independent of the hole heights, and (2) the hole heights themselves converge in distribution to a discrete Gaussian random vector. These results confirm general predictions about height fluctuations for tilings on multiply connected domains.

Temperleyan Domino Tilings with Holes

TL;DR

This work analyzes asymptotic height fluctuations for uniformly random domino tilings on multiply connected Temperleyan domains by combining Kenyon's contour-integral moment formulas with theta-function methods on the doubled domain . It proves that the centered height fluctuation, after removing a random harmonic correction from inner-hole boundary data, converges in moments to a Gaussian free field on , while the hole boundary heights converge to a centered discrete Gaussian with shift and scale matrix , with the two components becoming asymptotically independent. The discrete Gaussian arises from random hole boundary data, and the results confirm predictions for multiply connected tilings and suggest universality of discrete Gaussian contributions in such models. The approach builds a bridge between dimer model fluctuations and compactified free-field behavior via the theta-function framework on the associated Riemann surface, extending the understanding of conformal invariance and universality beyond simply connected domains.

Abstract

We analyze asymptotic height function fluctuations in uniformly random domino tiling models on multiply connected Temperleyan domains. Starting from asymptotic formulas derived by Kenyon [arXiv:math-ph/9910002v1], we show that (1) the difference of the centered height function and a harmonic function with boundary values given by the (random) centered hole heights converges in the sense of moments to a Gaussian free field, which is independent of the hole heights, and (2) the hole heights themselves converge in distribution to a discrete Gaussian random vector. These results confirm general predictions about height fluctuations for tilings on multiply connected domains.

Paper Structure

This paper contains 11 sections, 6 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Results
  4. Acknowledgments
  5. Preliminaries
  6. Results of Ken99, functions $F_+$ and $F_-$
  7. The double of a planar domain, theta functions, and prime forms
  8. Extensions and properties of $F_+$ and $F_-$
  9. Computing the joint moments
  10. A formula for $F_+$ and $F_-$
  11. Completing the proof

Key Result

Theorem 1.1

Fix pairwise distinct points $z_1,\dots, z_K \in U$. For each $\epsilon$ choose lattice points $z_1^{(\epsilon)},\dots, z_K^{(\epsilon)}$ of $P_\epsilon$, such that $z_i^{(\epsilon)}$ is within $O(\epsilon)$ of $z_i$. The joint moments of $\tilde{h}$ converges to those of the Gaussian free field in The summation in eqn:joint_moment_in is over all pairings, or partitions of $\{1,\dots, K\}$ into s

Figures (2)

  • Figure 1: The $M$ curve obtained as double of the planar domain $U$ with marked points $d_0,\dots,d_g$, in the case when $U$ is the upper half plane with $g$ circular holes cut out.
  • Figure 2: Integration paths for height moments. Some points $z_i$ may be on boundary circles.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • proof : proof of Theorem \ref{['thm:intro_main1']}
  • ...and 1 more