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1 September 2018 On the maximum of the CβE field
Reda Chhaibi, Thomas Madaule, Joseph Najnudel
Duke Math. J. 167(12): 2243-2345 (1 September 2018). DOI: 10.1215/00127094-2018-0016

Abstract

In this article, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according to the circular beta ensemble (CβE). More precisely, assuming that Xn is this characteristic polynomial and U is the unit circle, we prove that

sup zUlogXn(z)=2β(logn34loglogn+O(1)), as well as an analogous statement for the imaginary part. The notation O(1) means that the corresponding family of random variables, indexed by n, is tight. This answers a conjecture of Fyodorov, Hiary, and Keating, originally formulated for the β=2 case, which corresponds to the circular unitary ensemble (CUE) field.

Citation

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Reda Chhaibi. Thomas Madaule. Joseph Najnudel. "On the maximum of the CβE field." Duke Math. J. 167 (12) 2243 - 2345, 1 September 2018. https://doi.org/10.1215/00127094-2018-0016

Information

Received: 4 January 2017; Revised: 6 March 2018; Published: 1 September 2018
First available in Project Euclid: 10 August 2018

zbMATH: 06966872
MathSciNet: MR3848391
Digital Object Identifier: 10.1215/00127094-2018-0016

Subjects:
Primary: 60B20
Secondary: 60G70

Keywords: circular β ensembles , extremas of log-correlated fields , hierarchical structure , orthogonal polynomials on the unit circle (OPUCs) , Random matrix theory

Rights: Copyright © 2018 Duke University Press

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Vol.167 • No. 12 • 1 September 2018
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