Dynamical Correlations for Circular Ensembles of Random Matrices

TL;DR

This paper analyzes the dynamical correlations in circular ensembles of random matrices using Brownian motion models, revealing quaternion determinant structures similar to hermitian models.

Contribution

It extends the analysis of Dyson's circular Brownian motion models to include symmetric, self-dual quaternion, and antisymmetric hermitian initial conditions, deriving explicit correlation functions.

Findings

Dynamical correlation functions are expressed as quaternion determinants.

The results generalize known hermitian model structures to circular ensembles.

The analysis covers arbitrary numbers of particles and times.

Abstract

Circular Brownian motion models of random matrices were introduced by Dyson and describe the parametric eigenparameter correlations of unitary random matrices. For symmetric unitary, self-dual quaternion unitary and an analogue of antisymmetric hermitian matrix initial conditions, Brownian dynamics toward the unitary symmetry is analyzed. The dynamical correlation functions of arbitrary number of Brownian particles at arbitrary number of times are shown to be written in the forms of quaternion determinants, similarly as in the case of hermitian random matrix models.