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 Kullback–Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. Killip and Simon [24] have given a revival interest to this subject by showing a quite surprising sum rule for measures dominating the semicircular distribution on [−2,2][−2,2]. This sum rule includes a contribution of the atomic part of the measure away from [−2,2][−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.