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2021 Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications
Gaultier Lambert
Electron. J. Probab. 26: 1-33 (2021). DOI: 10.1214/20-EJP559

Abstract

We give a simple proof of a central limit theorem for linear statistics of the circular $\beta $-ensembles which is valid at almost microscopic scales for functions of class $C^{3}$. Using a coupling introduced by Valkò and Viràg [48], we deduce a central limit theorem for the Sine$_{\beta }$ processes. We also discuss connections between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of [37], we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the circular $\beta $-ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This establishes that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions coming from log-correlated Gaussian field theory.

Citation

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Gaultier Lambert. "Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications." Electron. J. Probab. 26 1 - 33, 2021. https://doi.org/10.1214/20-EJP559

Information

Received: 9 April 2020; Accepted: 18 November 2020; Published: 2021
First available in Project Euclid: 7 January 2021

Digital Object Identifier: 10.1214/20-EJP559

Subjects:
Primary: 60B20, 60F05, 60G20

JOURNAL ARTICLE
33 PAGES


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Vol.26 • 2021
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