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Sum rules via large deviations Rouault, Alain


In the theory of orthogonal polynomials, some 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. I will give a short historical introduction, from Szegö until Killip and Simon and descendants. These last authors proved in 2003 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]. It is possible to recover this sum rule and to establish new ones by using large deviations of spectral measures in beta-ensembles. These formulas include a contribution of ouliers. The method is robust enough to allow extensions to matrix-valued measures and to measures on the unit circle. (joint work with F. Gamboa and J. Nagel)

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