Dimension Results for the Spectral Measure of the Circular Beta Ensembles


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.

Annals of Applied Probability
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