Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

TL;DR

This paper investigates the recurrence times of eigenvalues of powers of large random unitary matrices, using Toeplitz determinants and loop equations to derive asymptotic behaviors and connect to quantum measurement phenomena.

Contribution

It introduces a novel approach linking Toeplitz determinants, loop equations, and eigenvalue recurrence times for powers of random unitary matrices, extending Widom's formula.

Findings

Eigenvalues of matrix powers tend to cluster near 1 at certain times.

The first return time distribution converges to an exponential distribution as N grows.

Numerical simulations support the theoretical asymptotic results.

Abstract

The purpose of this article is to study the eigenvalues $u_1^{\, t}=e^{it\theta_1},\dots,u_N^{\,t}=e^{it\theta_N}$ of $U^t$ where $U$ is a large $N\times N$ random unitary matrix and $t>0$. In particular we are interested in the typical times $t$ for which all the eigenvalues are simultaneously close to $1$ in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first orders of the large $N$ asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge towards an exponential distribution when $N$ is large. Numeric simulations are provided along the paper to illustrate the results.