Jan Nagel: Sum rules via large deviations

Abstract: We show a large deviation principle for the weighted spectral measure of random matrices corresponding to a general potential. Unlike for the empirical eigenvalue distribution, the speed reduces to n and the rate function contains a contribution of eigenvalues outside of the limit support. As an application, this large deviation principle yields a probabilistic proof of the celebrated Killip-Simon sum rule: a remarkable relation between the entries of a Jacobi-operator and its spectral measure. The talk is based on joint works with Fabrice Gamboa and Alain Rouault.