Central Limit Theorems for the Brownian motion on large unitary groups

TL;DR

This paper investigates the large N behavior of linear combinations of entries from Brownian motion on unitary groups, showing convergence to Gaussian processes across different scales and initial conditions.

Contribution

It provides a unified proof of the Gaussian limit for linear combinations of unitary Brownian motion entries, extending previous results to various scales and initial distributions.

Findings

Linear combinations converge to Gaussian processes as N grows large.

The convergence holds under various time scales and initial distributions.

A short proof of Gaussianity for Haar-distributed unitary matrices is provided.

Abstract

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.