Second order asymptotics for matrix models

We study several-matrix models and show that when the potential is convex and a small perturbation of the Gaussian potential, the first order correction to the free energy can be expressed as a generating function for the enumeration of maps of genus one. In order to do that, we prove a central limit theorem for traces of words of the weakly interacting random matrices defined by these matrix models and show that the variance is a generating function for the number of planar maps with two vertices with prescribed colored edges.Comment: Published in at http://dx.doi.org/10.1214/009117907000000141 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Modèles matriciels unitaires

National audienceNous présenterons dans cet exposé un travail réalisé avec Benoît Collins et Alice Guionnet sur le spectre de matrices unitaires en grande dimension

Asymptotics of unitary and othogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large $N$ limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the othogonal group.Comment: 41 pages, important modifications, new section about orthogonal integral

Modèles matriciels unitaires

National audienceNous présenterons dans cet exposé un travail réalisé avec Benoît Collins et Alice Guionnet sur le spectre de matrices unitaires en grande dimension