Central limit theorems for the brownian motion on large unitary groups

Benaych-Georges, Florent

Bulletin de la Société Mathématique de France, Tome 139 (2011) no. 4, pp. 593-610.

Dans cet article, on considère la loi limite, lorsque $n$ tend vers l’infini, de combinaisons linéaires des coefficients d’un mouvement Brownien sur le groupe des matrices unitaires $n\times n$. On prouve que le processus d’une telle combinaison linéaire converge vers un processus gaussien. Différentes échelles de temps et différentes lois initiales sont considérées, donnant lieu à plusieurs processus limites, liés à la construction géométrique du mouvement Brownien unitaire. En application, on propose une preuve très courte du caractère asymptotiquement gaussien des coefficients d’une matrice unitaire distribuée selon la mesure de Haar, un résultat déjà prouvé par Diaconis et al.

In this paper, we are concerned with the large $n$ limit of the distributions of linear combinations of the entries of a Brownian motion on the group of $n\times n$ unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic Gaussian feature of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al.

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