On asymptotics of large Haar distributed unitary matrices

TL;DR

This paper proves the asymptotic normality and independence of traces of powers of large Haar unitary matrices, and shows that renormalized truncated Haar unitaries converge to a Gaussian matrix, using elementary methods.

Contribution

It establishes new elementary proofs for the asymptotic behavior of traces of Haar unitaries and their convergence to Gaussian matrices.

Findings

Traces of powers of large Haar unitaries are asymptotically normal and independent.

Renormalized truncated Haar unitaries converge to Gaussian random matrices.

Elementary methods suffice for proving these asymptotic properties.

Abstract

Let $U_n$ be an $n \times n$ Haar unitary matrix. In this paper, the asymptotic normality and independence of $\Tr U_n, \Tr U_n^2, ..., \Tr U_n^k$ are shown by using elementary methods. More generally, it is shown that the renormalized truncated Haar unitaries converge to a Gaussian random matrix in distribution.