### 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 language | English |
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Pages (from-to) | 3058-3085 |

Number of pages | 28 |

Journal | Journal of Mathematical Physics |

Volume | 45 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2004 Aug 1 |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 45, no. 8, pp. 3058-3085. https://doi.org/10.1063/1.1765215

}

TY - JOUR

T1 - Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

AU - Katori, Makoto

AU - Tanemura, Hideki

PY - 2004/8/1

Y1 - 2004/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=4544227951&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544227951&partnerID=8YFLogxK

U2 - 10.1063/1.1765215

DO - 10.1063/1.1765215

M3 - Article

AN - SCOPUS:4544227951

VL - 45

SP - 3058

EP - 3085

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 8

ER -