We study the dimension properties of the spectral measure of the circular β-ensembles. For it was previously shown by Simon that the spectral measure is almost surely singular continuous with respect to Lebesgue measure on and the dimension of its support is . 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.
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.
Tom Alberts. Raoul Normand. "Dimension results for the spectral measure of the circular β ensembles." Ann. Appl. Probab. 32 (6) 4642 - 4680, December 2022. https://doi.org/10.1214/22-AAP1798