The Laguerre Unitary Process

TL;DR

This paper introduces a new matrix-valued stochastic process derived from the Laguerre Unitary Ensemble, exhibiting determinantal point process properties and connecting to Dyson Brownian Motion in a long-term limit.

Contribution

It defines a novel matrix process with stationary increments, provides explicit correlation kernel expressions, and links it to Dyson Brownian Motion through scaling limits.

Findings

Eigenvalues form a determinantal point process

Explicit correlation kernel in terms of Laguerre polynomials

Kernel converges to Dyson Brownian Motion in long time limit

Abstract

We define a new matrix-valued stochastic process with independent stationary increments from the Laguerre Unitary Ensemble, which in a certain sense may be considered a matrix generalisation of the gamma process. We show that eigenvalues of this matrix-valued process forms a spatiotemporal determinantal point process and give an explicit expression for the correlation kernel in terms of Laguerre polynomials. Furthermore, we show that in an appropriate long time scaling limit, this correlation kernel becomes identical to that of Dyson Brownian Motion.