Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes

TL;DR

This paper establishes a time-shift equivalence between noncolliding Brownian motions starting from GUE eigenvalues and from the origin, and derives new determinantal formulas for characteristic polynomial averages in related random matrix ensembles.

Contribution

It demonstrates a novel equivalence in noncolliding diffusion processes and derives explicit determinantal formulas for characteristic polynomial averages in multiple random matrix ensembles.

Findings

Time shift relates noncolliding BM from GUE eigenvalues to origin

Determinantal expressions for characteristic polynomial averages

Extension to noncolliding squared Bessel processes and related ensembles

Abstract

We consider the noncolliding Brownian motion (BM) with $N$ particles starting from the eigenvalue distribution of Gaussian unitary ensemble (GUE) of $N \times N$ Hermitian random matrices with variance $\sigma^2$. We prove that this process is equivalent with the time shift $t \to t+\sigma^2$ of the noncolliding BM starting from the configuration in which all $N$ particles are put at the origin. In order to demonstrate nontriviality of such equivalence for determinantal processes, we show that, even from its special consequence, determinantal expressions are derived for the ensemble averages of products of characteristic polynomials of random matrices in GUE. Another determinantal process, noncolliding squared Bessel process with index $\nu >-1$, is also studied in parallel with the noncolliding BM and corresponding results for characteristic polynomials are given for random matrices in the chiral GUE as well as in the Gaussian ensembles of class C and class D.