Sum rules and large deviations for spectral matrix measures

Abstract

In the paradigm of random matrices, one of the most classical object under study is the empirical spectral distribution. This random measure is the uniform distribution supported by the eigenvalues of the random matrix. In this paper, we give large deviation theorems for another popular object built on Hermitian random matrices: the spectral measure. This last probability measure is a random weighted version of the empirical spectral distribution. The weights involve the eigenvectors of the random matrix. We have previously studied the large deviations of the spectral measure in the case of scalar weights. Here, we will focus on matrix valued weights. Our probabilistic results lead to deterministic ones called “sum rules” in spectral theory. A sum rule relative to a reference measure on $\mathbb{R}$ is a relationship between the reversed Kullback–Leibler divergence of a positive measure on $\mathbb{R}$ and some non-linear functional built on spectral elements related to this measure. By using only probabilistic tools of large deviations, we extend the sum rules to the case of Hermitian matrix-valued measures.