Gaussian Unitary Ensemble in random lozenge tilings

TL;DR

This paper proves that the local behavior of large random lozenge tilings near certain boundary configurations follows the Gaussian Unitary Ensemble (GUE) statistics, revealing a universal connection to random matrix theory.

Contribution

It establishes a universality result linking lozenge tilings with GUE-corners process under specific boundary conditions, extending understanding of tiling fluctuations.

Findings

GUE-corners process describes local tiling asymptotics

Height function fluctuations are smaller than N^{1/2}

Universality holds for domains with boundary segments inclined at 120 degrees

Abstract

This paper establishes a universality result for scaling limits of uniformly random lozenge tilings of large domains. We prove that whenever a boundary of the domain has three adjacent straight segments inclined under 120 degrees to each other, the asymptotics of tilings near the middle segment is described by the GUE--corners process of random matrix theory. An important step in our argument is to show that fluctuations of the height function of random tilings on essentially arbitrary simply-connected domains of diameter $N$ have smaller magnitude than $N^{1/2}$.