Central Limit Theorems for the Brownian motion on large unitary groups

Florent Benaych-Georges (1, 2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

In this paper, we are concerned with the large N limit of linear combinations of the entries of a Brownian motion on the group of N by 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 distribution are concerned, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a quite short proof of the asymptotic Gaussian feature of the linear combinations of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al.

Mots clés: Unitary Brownian Motion; Heat Kernel; Random Matrices; Central Limit Theorem; Haar measure

Domaines: Probabilités [math.PR]

Florent Benaych-Georges. Central Limit Theorems for the Brownian motion on large unitary groups. 2009. ⟨hal-00374929v4⟩