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15 September 2019 Fourier transform on high-dimensional unitary groups with applications to random tilings
Alexey Bufetov, Vadim Gorin
Duke Math. J. 168(13): 2559-2649 (15 September 2019). DOI: 10.1215/00127094-2019-0023


A combination of direct and inverse Fourier transforms on the unitary group U(N) identifies normalized characters with probability measures on N-tuples of integers. We develop the N version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with the law of large numbers and the central limit theorem for global behavior of corresponding random N-tuples.

As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon–Okounkov conjecture (which predicts asymptotic Gaussianity and the exact form of the covariance) for a family of non-simply-connected polygons.

Another application is a central limit theorem for the U(N) quantum random walk with random initial data.


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Alexey Bufetov. Vadim Gorin. "Fourier transform on high-dimensional unitary groups with applications to random tilings." Duke Math. J. 168 (13) 2559 - 2649, 15 September 2019.


Received: 7 January 2018; Revised: 29 January 2019; Published: 15 September 2019
First available in Project Euclid: 7 September 2019

zbMATH: 07131294
MathSciNet: MR4007600
Digital Object Identifier: 10.1215/00127094-2019-0023

Primary: 60B15
Secondary: 22E65 , 60K35

Keywords: Asymptotic representation theory , noncommutative Fourier transform , random tilings

Rights: Copyright © 2019 Duke University Press


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Vol.168 • No. 13 • 15 September 2019
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