Multiplying unitary random matrices - universality and spectral properties

TL;DR

This paper analyzes the spectral properties of large products of random unitary matrices generated by Hermitian matrices, revealing universal eigenvalue density behavior and critical phenomena near specific spectral points.

Contribution

It provides a large N limit calculation of eigenvalue density for products of random unitary matrices, demonstrating universality and identifying critical behavior near eigenvalue spectrum edges.

Findings

Eigenvalue density depends only on the second moment of the generator.

Universality of spectral distribution in the large N limit.

Indications of critical behavior with eigenvalue spacing scaling near $ heta=\pi$.

Abstract

In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a hamiltonian random in time. We find that the result is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behavior (eigenvalue spacing scaling like $1/N^{3/4}$) close to $\theta=\pi$ for a specific critical evolution time $t_c$.