Kinetic Dyson Brownian motion

TL;DR

This paper investigates the spectral behavior of kinetic Brownian motion on Hermitian matrices, revealing eigenvalue persistence, the Markovian nature of eigenvalues only in 2D, and convergence to Dyson Brownian motion in large-scale limits.

Contribution

It demonstrates the eigenvalue dynamics of kinetic Brownian motion, showing eigenvalues remain distinct, and characterizes the Markovian property of eigenvalues only in two dimensions.

Findings

Eigenvalues stay distinct for all times.

Eigenvalues form a Markov process only when dimension d=2.

Eigenvalues converge to Dyson Brownian motion in large-scale limits.

Abstract

We study the spectrum of the kinetic Brownian motion in the space of $d\times d$ Hermitian matrices, $d\geq2$. We show that the eigenvalues stay distinct for all times, and that the process $\Lambda$ of eigenvalues is a kinetic diffusion (i.e. the pair $(\Lambda,\dot\Lambda)$ of $\Lambda$ and its derivative is Markovian) if and only if $d=2$. In the large scale and large time limit, we show that $\Lambda$ converges to the usual (Markovian) Dyson Brownian motion under suitable normalisation, regardless of the dimension.