Sum rules and large deviations for spectral matrix measures

TL;DR

This paper extends sum rules relating spectral measures and divergence functionals from scalar to matrix-valued cases using large deviations, recovering known results and deriving new ones for matrix semicircle and Marchenko-Pastur laws.

Contribution

It introduces a probabilistic approach to extend sum rules to Hermitian matrix-valued measures, including new results for matrix Marchenko-Pastur law.

Findings

Extended sum rules to matrix-valued spectral measures.

Recovered known scalar results for matrix semicircle law.

Derived new sum rule for matrix Marchenko-Pastur law.

Abstract

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.