A diffusive matrix model for invariant $\beta$-ensembles

TL;DR

This paper introduces a new diffusive matrix model that converges to the $eta$-Dyson Brownian motion for all $eta$ in [0,2], providing an explicit, invariant construction of $eta$-ensembles with detailed eigenvector dynamics.

Contribution

It presents a novel diffusive matrix model for $eta$-ensembles that is invariant under classical groups and describes the eigenvector behavior during eigenvalue collisions.

Findings

Model converges to $eta$-Dyson Brownian motion for all $eta ext{ in }[0,2]$

Eigenvector fluctuations are rapid and uniform during eigenvalue collisions when $eta<1$

Provides explicit construction of invariant $eta$-ensembles

Abstract

We define a new diffusive matrix model converging towards the $\beta$-Dyson Brownian motion for all $\beta\in [0,2]$ that provides an explicit construction of $\beta$-ensembles of random matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when $\beta< 1$ and that two eigenvalues collide, the eigenvectors of these two colliding eigenvalues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues.