Infinite Products of Large Random Matrices and Matrix-valued Diffusion

TL;DR

This paper extends random matrix theory to analyze spectral properties of large matrix products, revealing phase transitions in the spectrum and introducing a matrix-valued diffusion process, supported by analytical and numerical results.

Contribution

It introduces a novel diagrammatic approach for spectral analysis of infinite matrix products and defines a matrix-valued diffusion process with observed phase transitions.

Findings

Identification of topological phase transition in spectrum after critical diffusion time

Discovery of localization-delocalization transition in a specific matrix product ensemble

Validation of analytical results through numerical simulations

Abstract

We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large hermitian matrices. The infinite product case allows us to define a natural matrix-valued multiplicative diffusion process. In both cases of hermitian and complex matrices, we observe an emergence of "topological phase transition" in the spectrum, after some critical diffusion time $\tau_{\rm crit}$ is reached. In the case of the particular product of two hermitian ensembles, we observe also an unusual localization-delocalization phase transition in the spectrum of the considered ensemble. We verify the analytical formulae obtained in this work by numerical simulation.