Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

Makoto Katori, Hideki Tanemura

Research output: Contribution to journalArticle

72 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)3058-3085
Number of pages28
JournalJournal of Mathematical Physics
Volume45
Issue number8
DOIs
Publication statusPublished - 2004 Aug 1
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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