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 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