Central limit theorems for the brownian motion on large unitary groups

Florent Benaych-Georges

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 4, page 593-610
  • ISSN: 0037-9484

Abstract

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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 × 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 Diaconiset al.

How to cite

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Benaych-Georges, Florent. "Central limit theorems for the brownian motion on large unitary groups." Bulletin de la Société Mathématique de France 139.4 (2011): 593-610. <http://eudml.org/doc/272606>.

@article{Benaych2011,
abstract = {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 Diaconiset al.},
author = {Benaych-Georges, Florent},
journal = {Bulletin de la Société Mathématique de France},
keywords = {unitary brownian motion; heat kernel; random matrices; central limit theorem; Haar measure},
language = {eng},
number = {4},
pages = {593-610},
publisher = {Société mathématique de France},
title = {Central limit theorems for the brownian motion on large unitary groups},
url = {http://eudml.org/doc/272606},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Benaych-Georges, Florent
TI - Central limit theorems for the brownian motion on large unitary groups
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 4
SP - 593
EP - 610
AB - 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 Diaconiset al.
LA - eng
KW - unitary brownian motion; heat kernel; random matrices; central limit theorem; Haar measure
UR - http://eudml.org/doc/272606
ER -

References

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