Powers of large random unitary matrices and Toeplitz determinants

TL;DR

This paper investigates the asymptotic distribution of traces of powers of large random unitary matrices, revealing their convergence to independent complex normals through Toeplitz determinant analysis.

Contribution

It establishes a strong Szeg"o limit theorem for Toeplitz determinants with n-dependent symbols, connecting it to the behavior of traces of random unitary matrices.

Findings

Traces of powers of large random unitary matrices converge to independent complex normals.

The convergence holds for multiple powers with normalization by the square root of the minimum of the power and matrix size.

A new strong Szeg"o limit theorem for Toeplitz determinants with n-dependent symbols is proved.

Abstract

We study the limiting behavior of $\Tr U^{k(n)}$, where $U$ is a $n\times n$ random unitary matrix and $k(n)$ is a natural number that may vary with $n$ in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szeg\"o limit theorem for Toeplitz determinants associated to symbols depending on $n$ in a particular way. As a consequence to this result, we find that for each fixed $m\in \N$, the random variables $ \Tr U^{k_j(n)}/\sqrt{\min(k_j(n),n)}$, $j=1,..., m$, converge to independent standard complex normals.