# GUE corners limit of q-distributed lozenge tilings

### 2017/03/21

(with

Sevak Mkrtchyan)

*Electronic Journal of Probability, Volume 22 (2017), paper no. 101, 24 pp.* •

arXiv:1703.07503 [math.PR]

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.

Region where one sees the GUE statistics