Lozenge tilings with free boundaries

TL;DR

This paper investigates lozenge tilings with free boundaries, revealing that their local statistics near the boundary match the GUE-corners process and establishing a limit shape consistent with symmetric plane partitions.

Contribution

It introduces a new model of lozenge tilings with free boundaries and proves the emergence of GUE-corners statistics and a limit shape, extending known results to this setting.

Findings

GUE-corners process describes boundary lozenge positions

Existence of a symmetric limit shape for the height function

Boundary behavior matches fixed-boundary hexagon results

Abstract

We study lozenge tilings of a domain with partially free boundary. In particular, we consider a trapezoidal domain (half hexagon), s.t. the horizontal lozenges on the long side can intersect it anywhere to protrude halfway across. We show that the positions of the horizontal lozenges near the opposite flat vertical boundary have the same joint distribution as the eigenvalues from a Gaussian Unitary Ensemble (the GUE-corners/minors process). We also prove the existence of a limit shape of the height function, which is also a vertically symmetric plane partition. Both behaviors are shown to coincide with those of the corresponding doubled fixed-boundary hexagonal domain. We also consider domains where the different sides converge to $\infty$ at different rates and recover again the GUE-corners process near the boundary.