Sum rules and large deviations for spectral matrix measures

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

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 TUM - Technische Universität Munchen - Technical University Munich - Université Technique de Munich
3 LMV - Laboratoire de Mathématiques de Versailles

A sum rule relative to a reference measure on R is a relationship between the reversed Kullback-Leibler divergence of a positive measure on R and some non-linear functional built on spectral elements related to this measure (see for example Killip and Simon 2003). In this paper, using only probabilistic tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover the earlier result of Damanik, Killip and Simon (2010) when the reference measure is the (matrix-valued) semicircle law and obtain a new sum rule when the reference measure is the (matrix-valued) Marchenko-Pastur law.

Mots clés: Sum rules; block Jacobi matrix; matrix orthogonal polynomials; matrix-valued spectral measures; large deviations; random matrices

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]

Fabrice Gamboa, Jan Nagel, Alain Rouault. Sum rules and large deviations for spectral matrix measures. Bernoulli, 2019, 25 (1), pp.712-741. ⟨10.3150/17-BEJ1003⟩. ⟨hal-01264137v2⟩