Open Collections

Sum rules via large deviations

Banff International Research Station for Mathematical Innovation and Discovery

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)

Mathematics; Probability theory and stochastic processes; Statistical mechanics, structure of matter

video/mp4

Author affiliation: Université de Versailles-Saint-Quentin

2017-02-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59460

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/