Asymptotics for the number of domino tilings of L-shaped Aztec domains

TL;DR

<p><b>Problem and approach:</b> The paper analyzes the weighted number of domino tilings of L-shaped perturbations of the Aztec diamond, focusing on the asymptotics of the generating functions F_N^{m,k}(a;ε) as N oty and m,k scale with N. <b>Method:</b> The authors express ratio identities via a 2×2 Riemann–Hilbert problem, perform a sequence of contour deformations, and construct global and local parametrices (Painlevé II, Hermite and Airy) to obtain uniform asymptotics across three regimes: almost maximal, critical, and large removed corners. <b>Contributions:</b> They derive uniform log-scale expansions for log F_N^{m,k+1}(a;ε) capturing phase transitions, establish Tracy–Widom fluctuations in the critical regime, and prove a birth of a cut phenomenon with explicit constants and matching to TW tails. <b>Impact:</b> The results provide a detailed, uniform description of phase transitions in weighted domino tilings on perturbed Aztec diamonds and connect tiling counts to integrable structures and random matrix theory via TW and Painlevé II analysis.</p>

Abstract

We obtain precise asymptotics for the weighted number of domino tilings of an L-shaped subset of the Aztec diamond, obtained by removing an approximate rectangle in a corner of the Aztec diamond. By tuning the size of the removed corner, we observe different types of asymptotics. For a small removed corner, the number of tilings is close to that of the full Aztec diamond. Enlarging the removed corner to a critical size, a phase transition described in terms of the Tracy-Widom distribution occurs. Further increasing the size of the removed region, we observe a sharp decrease of the number of tilings, until it is finally approximated by the number of tilings of two smaller disjoint Aztec diamonds. We obtain uniform asymptotics for the number of domino tilings which fully describe these transitions.

Figures (15)

• Figure 1: Left: a domino tiling of $A_{5}$. The north, south, east and west dominoes are shown in red, yellow, green and blue, respectively. Right: a domino tiling of $A_{300}$ chosen uniformly at random.
• Figure 2: Left: a domino tiling of $A_{7}^{5,2}$. Right: a domino tiling of $\tilde{A}_{7}^{5,2}$.
• Figure 3: Left: a domino tiling of $A_{300}^{225,10}$ taken uniformly at random. Right: a domino tiling of $A_{300}^{225,102}$ taken uniformly at random.
• Figure 4: Let $\log \mathcal{F}_{N}^{1}$ be the right-hand side of \ref{['lol35']} without the ${\cal O}$-term. The left and middle figures represent $N \mapsto N(\log F_N^{m,k+1}(a;\epsilon)-\log \mathcal{F}_{N}^{1})$ with $a=0.7845$, $k=3$, $m=\{0.7N\}$, $\mu = \mu_{N} = m/N\approx 0.7$, and with $\epsilon=1$ (left) and $\epsilon=0$ (middle). Here $\{x\}$ denotes the nearest integer to $x$ (in case $x$ is exactly halfway between two integers, then $\{x\}$ is the nearest even integer to $x$). Right: Let $\log \mathcal{F}_{N}^{2}$ be the right-hand side of \ref{['asymp main thm']} without the ${\cal O}$-term and without the $o(1)$-term. The right figure represents $N \mapsto \log F_N^{m,k+1}(a;\epsilon)-\log \mathcal{F}_{N}^{2}$ with $a=0.7845$, $m=\{0.7N\}$, $\mu = \mu_{N} = m/N\approx 0.7$, $k=\{0.25N\}$, $\kappa=\kappa_{N} = k/N \approx 0.25$, and $\epsilon=1$.
• Figure 5: Left: a domino tiling of $A_{300}^{225,191}$ taken uniformly at random. Right: a domino tiling of $A_{300}^{225,215}$ taken uniformly at random.

Theorems & Definitions (50)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Theorem 1.9
• Remark 1.10