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

Fabrice Gamboa (1, 2) , Jan Nagel (3) , Alain Rouault (4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 TUM - Technische Universität Munchen - Technical University Munich - Université Technique de Munich
4 LMV - Laboratoire de Mathématiques de Versailles

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.

Mots clés: Kullback-Leibler divergence; orthogonal polynomials; spectral measures; Sum rules; large deviations; random matrices 1

Domaines: Probabilités [math.PR]

Fabrice Gamboa, Jan Nagel, Alain Rouault. Sum rules and large deviations for spectral measures on the unit circle. Random Matrices: Theory and Applications, 2017, 6 (1), ⟨10.1142/S2010326317500058⟩. ⟨hal-01300205v6⟩