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

TL;DR

This paper derives detailed asymptotic formulas for domino tilings of L-shaped Aztec domains, revealing phase transitions and different regimes depending on the size of the removed corner.

Contribution

It provides the first precise asymptotic analysis of domino tilings for L-shaped Aztec domains, including phase transition phenomena.

Findings

Asymptotics match full Aztec diamond for small removals

Phase transition occurs at critical removal size, described by Tracy-Widom distribution

Number of tilings sharply decreases with larger removed regions

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.