Dimension Results for the Spectral Measure of the Circular Beta Ensembles

Abstract

We study the dimension properties of the spectral measure of the Circular β-Ensembles. For β≥2 it 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 1−2/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.

Publication
To appear in Ann. Appl. Prob.
Tom Alberts
Tom Alberts
Associate Professor of Mathematics
University of Utah
Raoul Normand
Raoul Normand
Clinical Assistant Professor of Mathematics
New York University - Courant Institute

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