math.PR

# Title:GUE corners limit of q-distributed lozenge tilings

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.
 Comments: 25 pages, 8 figures; v2: minor changes in introduction Subjects: Probability (math.PR); Mathematical Physics (math-ph); Combinatorics (math.CO); Quantum Algebra (math.QA) Cite as: arXiv:1703.07503 [math.PR] (or arXiv:1703.07503v2 [math.PR] for this version)

## Submission history

From: Leonid Petrov [view email]
[v1] Wed, 22 Mar 2017 03:13:13 UTC (544 KB)
[v2] Wed, 5 Apr 2017 02:09:52 UTC (545 KB)
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