GUE corners limit of q-distributed lozenge tilings

TL;DR

This paper establishes that as the parameter q approaches 1, the distribution of vertical lozenges in large q-weighted lozenge tilings converges to the eigenvalue distribution of GUE matrices, extending known results from the uniform case.

Contribution

It proves a GUE corners asymptotics result for q-distributed lozenge tilings, generalizing previous uniform case findings to include q-weighted models.

Findings

Vertical lozenges follow GUE eigenvalue distributions asymptotically.

Results extend GUE corners asymptotics to q-weighted tilings.

Non-universal constants are affected by the q-weighting even as q approaches 1.

Abstract

We study asymptotics of $q$-distributed random lozenge tilings of sawtooth domains (equivalently, of random interlacing integer arrays with fixed top row). Under the distribution we consider each tiling is weighted proportionally to $q^{\mathsf{vol}}$, where $\mathsf{vol}$ is the volume under the corresponding 3D stepped surface. We prove the following Interlacing Central Limit Theorem: as $q\rightarrow1$, the domain gets large, and the fixed top row approximates a given nonrandom profile, the vertical lozenges are distributed as the eigenvalues of a GUE random matrix and of its successive principal corners. Our results extend the GUE corners asymptotics for tilings of bounded polygonal domains previously known in the uniform (i.e., $q=1$) case. Even though $q$ goes to $1$, the presence of the $q$-weighting affects non-universal constants in our Central Limit Theorem.