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December 2022 Dimension results for the spectral measure of the circular β ensembles
Tom Alberts, Raoul Normand
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Ann. Appl. Probab. 32(6): 4642-4680 (December 2022). DOI: 10.1214/22-AAP1798


We study the dimension properties of the spectral measure of the circular β-ensembles. For β2 it was previously shown by Simon that the spectral measure is almost surely singular continuous with respect to Lebesgue measure on D and the dimension of its support is 12/β. We reprove this result with a combination of probabilistic techniques and the so-called Jitomirskaya–Last inequalities. Our method is simpler in nature and mostly self-contained, with an emphasis on the probabilistic aspects rather than the analytic. We also extend the method to prove a large deviations principle for norms involved in the Jitomirskaya–Last analysis.

Funding Statement

T. Alberts was partially supported by NSF Grants DMS-1715680, DMS-1811087, and Simons Collaboration Grant 351687.


We thank Bálint Virág for many helpful discussions, especially for suggesting the proof of Section 4.5.2. We also thank an anonymous referee for many helpful suggestions that lead to a much improved version of this paper.


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Tom Alberts. Raoul Normand. "Dimension results for the spectral measure of the circular β ensembles." Ann. Appl. Probab. 32 (6) 4642 - 4680, December 2022.


Received: 1 March 2020; Revised: 1 February 2021; Published: December 2022
First available in Project Euclid: 6 December 2022

Digital Object Identifier: 10.1214/22-AAP1798

Primary: 60B20
Secondary: 15B52 , 60F10

Keywords: circular-β ensemble , Hausdorff dimension , random matrices , spectral measure

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.32 • No. 6 • December 2022
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