Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Domino tilings of black-and-white Temperleyan cylinders

Dmitry Chelkak, Zachary Deiman

TL;DR

This work analyses domino tilings (the dimer model) on black-and-white Temperleyan cylinders, establishing that height fluctuations converge to the Gaussian Free Field on the limit domain $\Omega$ plus an independent discrete Gaussian component along the harmonic measure of the top boundary. The authors construct and study the limiting holomorphic limits $f_{\Omega}^{[\eta]}$ of dimer coupling functions $K^{-1}_{\Omega_\delta}$, showing these limits exist but are not conformally covariant, which drives the appearance of the instanton (discrete Gaussian) part. By expressing multi-point height correlations through the differential forms $\mathcal{A}_{n,\Omega}$ built from the $f_{\Omega}^{[s_j,s_k]}$, they derive a cubic relation between the second and third moments, parametrize all moments via elliptic data, and introduce model functions $f^{[\pm\pm]}_{\mu}$ using theta-functions to compute cumulants. The result is a precise description: the height field converges in distribution to $\pi^{-1/2}\mathrm{GFF}_{\Omega}$ plus an independent discrete Gaussian along the top boundary, with cumulants tied to the discrete Gaussian parameter $\mu$, thereby formalizing the occurrence of the discrete Gaussian component in a non-simply connected setting and extending the understanding of instanton contributions in doubly connected domains.

Abstract

We consider the dimer model in cylindrical domains $Ω_δ$ on square grids of mesh size $δ$ with two Temperleyan boundary components of different colors. Assuming that the $Ω_δ$ approximate a cylindrical domain $Ω$ as $δ\to 0$, we prove the convergence of height fluctuations to the Gaussian Free Field in $Ω$ plus an independent discrete Gaussian multiple of the harmonic measure of one of the boundary components. The limit of the dimer coupling functions on $Ω_δ$ is holomorphic in $Ω$ but not conformally covariant. Given this, we determine the limiting structure of height fluctuations from general principles rather than from explicit computations. In particular, our analysis justifies the inevitable appearance of the discrete Gaussian distribution in the doubly connected setup.

Domino tilings of black-and-white Temperleyan cylinders

TL;DR

This work analyses domino tilings (the dimer model) on black-and-white Temperleyan cylinders, establishing that height fluctuations converge to the Gaussian Free Field on the limit domain plus an independent discrete Gaussian component along the harmonic measure of the top boundary. The authors construct and study the limiting holomorphic limits of dimer coupling functions , showing these limits exist but are not conformally covariant, which drives the appearance of the instanton (discrete Gaussian) part. By expressing multi-point height correlations through the differential forms built from the , they derive a cubic relation between the second and third moments, parametrize all moments via elliptic data, and introduce model functions using theta-functions to compute cumulants. The result is a precise description: the height field converges in distribution to plus an independent discrete Gaussian along the top boundary, with cumulants tied to the discrete Gaussian parameter , thereby formalizing the occurrence of the discrete Gaussian component in a non-simply connected setting and extending the understanding of instanton contributions in doubly connected domains.

Abstract

We consider the dimer model in cylindrical domains on square grids of mesh size with two Temperleyan boundary components of different colors. Assuming that the approximate a cylindrical domain as , we prove the convergence of height fluctuations to the Gaussian Free Field in plus an independent discrete Gaussian multiple of the harmonic measure of one of the boundary components. The limit of the dimer coupling functions on is holomorphic in but not conformally covariant. Given this, we determine the limiting structure of height fluctuations from general principles rather than from explicit computations. In particular, our analysis justifies the inevitable appearance of the discrete Gaussian distribution in the doubly connected setup.
Paper Structure (16 sections, 18 theorems, 109 equations, 3 figures)

This paper contains 16 sections, 18 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Setup and convergence of the dimer coupling functions
  4. Convergence of height fluctuations
  5. Proofs of Theorem \ref{['Kdconvergencetheorem']} and Theorem \ref{['Hnconvergencetheorem']}
  6. Discrete holomorphicity and boundary conditions of dimer coupling functions
  7. Proof of Theorem \ref{['Kdconvergencetheorem']}
  8. Functions $f^{[\pm\pm]}_\Omega$ and their discrete counterparts
  9. Proof of Theorem \ref{['Hnconvergencetheorem']}
  10. Identification of the limit
  11. Uniqueness of solutions of Problem \ref{['bvp']}
  12. Differential forms $\mathcal{A}_n$ on $T^+$, their primitives $h_n$, and the sequence of moments $M_n$
  13. Cubic equation for the second and third moments
  14. Model functions $f^{[\pm\pm]}_\mu$
  15. Connected correlation functions of height fluctuations
  16. ...and 1 more sections

Key Result

Theorem 1.4

Fix $\mathfrak{w},\mathfrak{b}\in\{0,1\}$. Let $w_\delta\in W_\mathfrak{w}(\Omega_\delta)$, $b_\delta\in B_\mathfrak{b}(\Omega_\delta)$ be sequences of white and black vertices of types $W_\mathfrak{w}$ and $B_\mathfrak{b}$, respectively, such that $w_\delta\to z_1\in\Omega$ and $b_\delta\to z_2\in\ where $\eta_0:=1$ and $\eta_1:=i$. Moreover, this convergence is uniform provided that $z_1,z_2$ st

Figures (3)

  • Figure 1: Left: An example of a black-and-white Temperleyan cylinder $\Omega_\delta$. Of the black (resp., white) vertices, the circular nodes are of type $B_0$ (resp., $W_0$), and the square nodes are of type $B_1$ (resp., $W_1$). The boundary $\partial\Omega_\delta:=\partial_\textnormal{bot}\Omega_\delta\cup\partial_\textnormal{top}\Omega_\delta$ of $\Omega_\delta$ consists of the vertices marked with $\times$ which are not part of the domain itself. Top-right: A piece of the dual graph $\Omega_\delta^*$ with square faces of types $B_0,B_1,W_0,W_1$. Bottom-right: The Kasteleyn weights on edges of the corresponding piece of $\Omega_\delta$.
  • Figure 2: Left: Notation used in the proof of Proposition \ref{['uniformboundedness']}: a point $w\in\overline{\Omega}{}^{(3d)}$ and the regions $\Omega^{(d)}_{w}$, $\Omega_\textnormal{top}$ and $\Omega_\textnormal{bot}$ of $\Omega\smallsetminus \overline{B(w,d)}$. Note that $\overline{B(w,d)}\subset\Omega^{(d)}$. Right: A splitting of the white squares of $\delta\mathbb{Z}^2$ and the types of values ($\eta_b\mathbb{R}$ or $\mathbb{C}$) of t-white-holomorphic functions on this splitting; see also clrI.
  • Figure 3: The real locus of the elliptic curve defined by the cubic equation \ref{['cubicequation']} and parameterized by $z\mapsto (-\wp(z)-c_\mathfrak{m},\wp'(z))$ with $\omega_1=\mathfrak{m}=3\pi$ and $\omega_2=2\pi i$. The left component corresponds to $z\in\mathbb{R}/\omega_1\mathbb{Z}$ and the right one to $z-\frac{1}{2}\omega_2\in\mathbb{R}/\omega_1\mathbb{Z}$. Since $M_2\ge 0$, all possible pairs $(M_2,M_3)$ belong to the right component.

Theorems & Definitions (47)

  • Remark 1.1
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 2.1
  • Proposition 2.2
  • proof
  • ...and 37 more