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The Airy line ensemble at the rough-smooth boundary

Sunil Chhita, Duncan Dauvergne, Thomas Finn

Abstract

We study the rough-smooth boundary in the two-periodic Aztec diamond, a random domino tiling model exhibiting three types of macroscopic regions. We show that the height function at this boundary converges to an independent sum of an Airy surface and an i.i.d. noise field with fluctuations governed by the full-plane smooth phase. Going further, we prove convergence of a family of Temperleyan backbone paths to the Airy line ensemble. This gives the first convergence result for a family of undirected paths converging to the Airy line ensemble, as well as Airy convergence at a noisy boundary.

The Airy line ensemble at the rough-smooth boundary

Abstract

We study the rough-smooth boundary in the two-periodic Aztec diamond, a random domino tiling model exhibiting three types of macroscopic regions. We show that the height function at this boundary converges to an independent sum of an Airy surface and an i.i.d. noise field with fluctuations governed by the full-plane smooth phase. Going further, we prove convergence of a family of Temperleyan backbone paths to the Airy line ensemble. This gives the first convergence result for a family of undirected paths converging to the Airy line ensemble, as well as Airy convergence at a noisy boundary.
Paper Structure (44 sections, 76 theorems, 406 equations, 25 figures)

This paper contains 44 sections, 76 theorems, 406 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Previous work on the model
  3. Boundaries beyond tilings and vertex models
  4. Outline of the paper
  5. Definitions and main results
  6. The two-periodic Aztec diamond
  7. Height functions
  8. Temperley's bijection
  9. Main results
  10. Proof sketch and some auxiliary results
  11. Trees, forests, and Temperley's bijection
  12. A generalization of Temperley's bijection
  13. Temperley's bijection and the two-periodic Aztec diamond
  14. The smooth phase.
  15. Basic tools
  16. ...and 29 more sections

Key Result

Theorem 1

Fix $a \in (0, 1)$. Let $\mathcal{H}_n$ be the height function of the two-periodic Aztec diamond of size $n$, and let $H_n$ be the average of $\mathcal{H}_n$ in the smooth region. Let $[\cdot]_n:\mathbb{R}^2 \to \mathbb{R}^2$ be a scaling map which takes limiting coordinates to an $O(n^{2/3}) \times where $\mathcal{A}$ is the Airy surface and $X = (X_u, u \in F)$ is an i.i.d. vector, independent o

Figures (25)

  • Figure 1: Temperleyan paths for the two-periodic Aztec diamond ($n = 1200, a = 0.5$).
  • Figure 2: A simulation of a two-periodic Aztec diamond with $n=200$ and $a=0.7$ with the $a$-dominoes shaded darker than the $b$-dominoes. Note the underlying square grid is rotated by $\pi/4$, as is our coordinate convention in this paper.
  • Figure 3: The same simulation as in \ref{['fig:intro-tiling']}, with an overlay of south ($\mathcal{S}$) and north ($\mathcal{N}$) backbone paths from the bottom and top of the Aztec diamond respectively.
  • Figure 4: The two-periodic Aztec diamond of size $4$. The edges around each face labeled $a$ have weight $a$ while the edges around each face labeled $b$ have weight $1$.
  • Figure 5: The relationship between dimers and dominoes is shown on the figure on the left while the figure on the right shows the height function for this dimer covering.
  • ...and 20 more figures

Theorems & Definitions (147)

  • Theorem 1: Informal version of \ref{['T:main-1']}
  • Theorem 2: Informal version of \ref{['T:main-2']}
  • Definition 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.2
  • Remark 2.4
  • Remark 2.5
  • Conjecture 2.3
  • ...and 137 more