Electronic Journal of Probability

GUE corners limit of $q$-distributed lozenge tilings

Sevak Mkrtchyan and Leonid Petrov

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 101, 24 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

lozenge tilings interlacing volume measure central limit theorem Gaussian unitary ensemble

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Mkrtchyan, Sevak; Petrov, Leonid. GUE corners limit of $q$-distributed lozenge tilings. Electron. J. Probab. 22 (2017), paper no. 101, 24 pp. doi:10.1214/17-EJP112. https://projecteuclid.org/euclid.ejp/1511578856

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