Sum rules via large deviations

In the theory of orthogonal polynomials, sum rules are remarkable relationships between a functional defined on a subset of all probability measures involving the reverse KullbackLeibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. Killip and Simon (Killip and Simon (2003)) have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on [−2, 2]. This sum rule includes a contribution of the atomic part of the measure away from [−2, 2]. In this paper, we recover this sum rule by using probabilistic tools on random matrices. Furthermore, we obtain new (up to our knowledge) magic sum rules for the reverse Kullback-Leibler divergence with respect to the Marchenko-Pastur or Kesten-McKay distributions. As in the semicircular case, these formulas include a contribution of the atomic part appearing away from the support of the reference measure.