Transportation techniques for beta-ensembles

Prof. Alessio Figalli
ETH, Zurich
Thursday, June 1, 2017 - 14:00

 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.

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