Sum rules and large deviations for spectral measures on the unit circle

TL;DR

This paper investigates sum rules and large deviations for spectral measures on the unit circle, focusing on the Gross-Witten and Hua-Pickrell ensembles, revealing new sum rules and probabilistic interpretations in matrix models.

Contribution

It introduces new sum rules for spectral measures related to the Gross-Witten and Hua-Pickrell ensembles, extending the understanding of large deviations in these models.

Findings

Probabilistic interpretation of a Simon sum rule in the Gross-Witten ensemble.

New sum rule for deviations of the Hua-Pickrell ensemble's equilibrium measure.

Extension of results to matrix measures.

Abstract

This work is a companion paper of Gamboa, Nagel, Rouault (J. Funct. Anal. 2016). We continue to explore the connections between large deviations for random objects issued from random matrix theory and sum rules. Here, we are concerned essentially with measures on the unit circle whose support is an arc that is possibly proper. We particularly focus on two matrix models. The first one is the Gross-Witten ensemble. In the gapped regime we give a probabilistic interpretation of a Simon sum rule. The second matrix model is the Hua-Pickrell ensemble. Unlike the Gross-Witten ensemble the potential is here infinite at one point. Surprisingly, but as in the above mentioned paper, we obtain a completely new sum rule for the deviation to the equilibrium measure of the Hua-Pickrell ensemble. The extension to matrix measures is also studied.