Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems
Abstract
As an extension of the theory of Dyson's Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor's generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland-Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish-Chandra (Itzykson-Zuber) formula of integral over unitary group is established.
Cite
@article{arxiv.math-ph/0402061,
title = {Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems},
author = {Makoto Katori and Hideki Tanemura},
journal= {arXiv preprint arXiv:math-ph/0402061},
year = {2007}
}
Comments
LaTeX, 27 pages, 4 figures, v3: Minor corrections made for publication in J. Math. Phys