21ème journée EDP/Probas
Transport optimal et matrices aléatoires
Vendredi 10 mars 2017
Alice Guionnet et Alessio Figalli donneront des exposés sur le thème du transport optimal et des matrices aléatoires.
10h-12h Alice Guionnet (ENS Lyon) Introduction to Beta-ensembles
I will discuss diverse properties of Beta-ensembles which turns out to be central in the analysis of transport maps : large deviations, concentration of measure, loop equations analysis and central limit theorems.
14h15-16h15 Alessio Figalli (ETH Zürich) Transporation techniques for beta-ensembles
Optimal transport theory is an efficient tool to construct change of variables between probability densities. However, when it comes to the regularity of these maps, one cannot hope to obtain estimates that are uniform with respect to the dimension except in some very special cases (for instance, between uniformly log-concave densities). In random matrix theory the densities involved are pretty singular, so it seems hopeless to apply optimal transport theory in this context. However, ideas coming from optimal transport can still be used to construct approximate transport maps (i.e., maps which send a density onto another up to a small error) which enjoy regularity estimates that are uniform in the dimension. Such maps can then be used to show universality results for beta-ensembles.
La journée aura lieu dans l’amphithéâtre Hermite de l’Institut Henri Poincaré, 11 rue Pierre et Marie Curie, Paris 5ème (accès).