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

Makoto Katori, Hideki Tanemura

Research output: Contribution to journalArticle

69 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

Fingerprint

particle diffusion
stochastic processes
Particle System
Stochastic Processes
eigenvalues
Eigenvalue
Random Matrices
Symmetry
Brownian motion
Ensemble
symmetry
matrices
Bessel Process
Stochastic Calculus
meanders
Unitary group
calculus
Hermitian matrix
Diffusion Process
equivalence

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. / Katori, Makoto; Tanemura, Hideki.

In: Journal of Mathematical Physics, Vol. 45, No. 8, 01.08.2004, p. 3058-3085.

Research output: Contribution to journalArticle

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