Large deviations, moment problems and sum rules

Abstract In this course, we study the convergence of random probability measures
which are spectral measures of random matrices. These measures can be described in different ways, in terms of moments, orthogonal polynomials or by their spectral information. In order to prove large deviation statements, it can be helpful to switch between different descriptions.
It turns out that there is a very interesting and surprising connection between large deviations for random spectral measures and important identities in spectral theory called sum rules. We will study these
identities from a probabilistic point of view, by proving large deviation statements in different encodings of the measures. This yields a new interpretation of sum rules and a probabilistic strategy to derive
new sum rules. lecture_1 lecture_2 lecture_3 exercises