Sum rules via large deviations

Abstract

In this paper, a sum rule means a relationship between a functional defined on a subset of all probability measures on $\mathbb{R}$ involving the reverse Kullback-Leibler divergence with respect to a particular distribution and recursion coefficients related to the orthogonal polynomial construction. The first sum rule is the Szeg\H{o}-Verblunsky theorem (see Simon (2011) Theorem 1.8.6 p. 29) and concerns the case of trigonometrical polynomials on the torus. More recently, 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. Indeed, this sum rule includes a contribution of the atomic part of the measure away from the support of the circular law. In this paper, we recover this sum rules by using only large deviations 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 Pastur-Marchenko or Kesten-McKay distributions. As in the semicircular case, these formulas generally include a contribution of the atomic part appearing away from the support of the reference measure. From the point of view of large deviations, we obtain these formulas by showing a large deviation principle for the point mass measure concentrated on random eigenvalues of classical ensembles and combine it with a large deviation principle due to Gamboa Rouault (2011) for the recursion coefficients.