Eigenvalue processes of symmetric tridiagonal matrix-valued processes associated with Gaussian beta ensemble

TL;DR

This paper derives stochastic differential equations for eigenvalues of symmetric tridiagonal matrix processes linked to the Gaussian beta ensemble, revealing how eigenvalues of minors interlace and conditions for non-collision.

Contribution

It introduces a novel stochastic differential equation framework for eigenvalues of matrix-valued processes associated with GβE, including interlacing properties and collision avoidance conditions.

Findings

Eigenvalue processes satisfy specific stochastic differential equations.

Eigenvalues of principal minors interlace according to derived equations.

Conditions for eigenvalue non-collision depend on Bessel process dimensions.

Abstract

We consider the symmetric tridiagonal matrix-valued process associated with Gaussian beta ensemble (G$\beta$E) by putting independent Brownian motions and Bessel processes on the diagonal entries and upper (lower)-diagonal ones, respectively. Then, we derive the stochastic differential equations that the eigenvalue processes satisfy, and we show that eigenvalues of their (indexed) principal minor sub-matrices appear in the stochastic differential equations. By the Cauchy's interlacing argument for eigenvalues, we can characterize the sufficient condition that the eigenvalue processes never collide with each other almost surely, by the dimensions of the Bessel processes.