Characteristic Polynomials of Random Matrices and Noncolliding Diffusion Processes

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.

v2: AMS_LaTeX, 23 pages, no figure

References in corpus (7)

• Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
• Exact distribution of the maximal height of p vicious walkers
• Differential Equations for Dyson Processes
• Extremal statistics of curved growing interfaces in 1+1 dimensions
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices
• Zeros of Airy Function and Relaxation Process
• Average characteristic polynomials in the two-matrix model

Cited by in corpus (1)

• Determinantal process starting from an orthogonal symmetry is a Pfaffian process