Open Access
2017 GUE corners limit of $q$-distributed lozenge tilings
Sevak Mkrtchyan, Leonid Petrov
Electron. J. Probab. 22: 1-24 (2017). DOI: 10.1214/17-EJP112


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.


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Sevak Mkrtchyan. Leonid Petrov. "GUE corners limit of $q$-distributed lozenge tilings." Electron. J. Probab. 22 1 - 24, 2017.


Received: 5 May 2017; Accepted: 25 September 2017; Published: 2017
First available in Project Euclid: 25 November 2017

zbMATH: 06827078
MathSciNet: MR3733659
Digital Object Identifier: 10.1214/17-EJP112

Primary: 60B20 , 60C05

Keywords: central limit theorem , Gaussian unitary ensemble , interlacing , lozenge tilings , Volume measure

Vol.22 • 2017
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