Boundary dents, the arctic circle and the arctic ellipse

TL;DR

This paper investigates how the number of domino tilings of Aztec diamonds changes with boundary defects, revealing a connection to the inscribed circle and Delannoy numbers, and deducing the shape of the arctic curve.

Contribution

It establishes a link between boundary defects and asymptotic tiling ratios, showing the arctic curve is the inscribed circle in the scaling limit.

Findings

The ratio of tilings approaches a Delannoy number for fixed boundary defects.

When defects approach boundary points outside the inscribed circle, asymptotics match Delannoy numbers.

If defects cross the inscribed circle, asymptotics differ radically.

Abstract

The original motivation for this paper goes back to the mid-1990's, when James Propp was interested in natural situations when the number of domino tilings of a region increases if some of its unit squares are deleted. Guided in part by the intuition one gets from earlier work on parallels between the number of tilings of a region with holes and the 2D Coulomb energy of the corresponding system of electric charges, we consider Aztec diamond regions with unit square defects along two adjacent sides. We show that for large regions, if these defects are at fixed distances from a corner, the ratio between the number of domino tilings of the Aztec diamond with defects and the number of tilings of the entire Aztec diamond approaches a Delannoy number. When the locations of the defects are not fixed but instead approach given points on the boundary of the scaling limit $S$ (a square) of the Aztec diamonds, we prove that, provided the line segment connecting these points is outside the circle inscribed in $S$, this ratio has the same asymptotics as the Delannoy number corresponding to the locations of the defects; if the segment crosses the circle, the asymptotics is radically different. We use this to deduce (under the assumption that an arctic curve exists) that the arctic curve for domino tilings of Aztec diamonds is the circle inscribed in $S$. We also discuss counterparts of this phenomenon for lozenge tilings of hexagons.