13 research outputs found
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
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 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
On the large N limit of matrix integrals over the orthogonal group
We reexamine the large N limit of matrix integrals over the orthogonal group
O(N) and their relation with those pertaining to the unitary group U(N). We
prove that lim_{N to infty} N^{-2} \int DO exp N tr JO is half the
corresponding function in U(N), and a similar relation for lim_{N to infty}
\int DO exp N tr(A O B O^t), for A and B both symmetric or both skew symmetric.Comment: 12 page
Matrix models at low temperature
65 pagesIn this article we investigate the behavior of multi-matrix unitary invariant models under a potential when the inverse temperature becomes very large. We first prove, under mild hypothesis on the functionals that as soon at these potentials are "confining" at infinity, the sequence of spectral distribution of the matrices are tight when the dimension goes to infinity. Their limit points are solutions of Dyson-Schwinger's equations. Next we investigate a few specific models, most importantly the "strong single variable model" where is a sum of potentials in a single matrix and the "strong commutator model" where
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