Dimension results for the spectral measure of the circular β ensembles

Abstract

We study the dimension properties of the spectral measure of the circular β-ensembles. For $\mathit{\beta }\ge 2$ it was previously shown by Simon that the spectral measure is almost surely singular continuous with respect to Lebesgue measure on $\partial \mathbb{D}$ and the dimension of its support is $1-2/\mathit{\beta }$. 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.