Eigenvalues of the laguerre process as non-colliding squared bessel processes

Abstract

Let A(t) be a n × p matrix with independent standard complex Brownian entries and set M(t) = A(t)*A(t). This is a process version of the Laguerre ensemble and as such we shall refer to it as the Laguerre process. The purpose of this note is to remark that, assuming n ≥ p, the eigenvalues of M(t) evolve like p independent squared Bessel processes of dimension 2(n - p+1), conditioned (in the sense of Doob) never to collide. More precisely, the function h(x) = ∏i