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 $q\rightarrow 1$, 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.
Citation
Sevak Mkrtchyan. Leonid Petrov. "GUE corners limit of $q$-distributed lozenge tilings." Electron. J. Probab. 22 1 - 24, 2017. https://doi.org/10.1214/17-EJP112
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