Orbit measures, random matrix theory and interlaced determinantal processes

Defosseux, Manon

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 1, pp. 209-249.

Nous décrivons les liens unissant les représentations de groupes compacts et certains ensembles invariants de matrices aléatoires. Cet article porte plus particulièrement sur deux types d'ensembles invariants qui généralisent les ensembles gaussiens ou de Laguerre. Nous les étudions en considérant des convolutions ou des projections de probabilités invariantes sur des orbites adjointes de groupes de Lie compacts. Par approximation semi-classique, ces mesures sont décrites par des produits tensoriels ou des restrictions de représentations. Nous montrons qu'une large classe d'entre elles sont déterminantales.

A connection between representation of compact groups and some invariant ensembles of hermitian matrices is described. We focus on two types of invariant ensembles which extend the gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal.

Classification : 15A52, 17B10
Mots clés : random matrix, determinantal process, interlaced configuration, Gelfand Tsetlin polytope, cristal graph, minor process, rank one perturbation

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