### Abstract

Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov–de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type [Formula presented] in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.

Original language | English |
---|---|

Number of pages | 1 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 68 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2003 Jan 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*68*(2). https://doi.org/10.1103/PhysRevE.68.021112

**Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov–de Gennes random matrices.** / Katori, Makoto; Tanemura, Hideki; Nagao, Taro; Komatsuda, Naoaki.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 68, no. 2. https://doi.org/10.1103/PhysRevE.68.021112

}

TY - JOUR

T1 - Vicious walks with a wall, noncolliding meanders, and chiral and Bogoliubov–de Gennes random matrices

AU - Katori, Makoto

AU - Tanemura, Hideki

AU - Nagao, Taro

AU - Komatsuda, Naoaki

PY - 2003/1/1

Y1 - 2003/1/1

N2 - Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov–de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type [Formula presented] in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.

AB - Spatially and temporally inhomogeneous evolution of one-dimensional vicious walkers with wall restriction is studied. We show that its continuum version is equivalent with a noncolliding system of stochastic processes called Brownian meanders. Here the Brownian meander is a temporally inhomogeneous process introduced by Yor as a transform of the Bessel process that is the motion of radial coordinate of the three-dimensional Brownian motion represented in spherical coordinates. It is proved that the spatial distribution of vicious walkers with a wall at the origin can be described by the eigenvalue statistics of Gaussian ensembles of Bogoliubov–de Gennes Hamiltonians of the mean-field theory of superconductivity, which have a particle-hole symmetry. We report that a time evolution of the present stochastic process is fully characterized by the change of symmetry classes from type C to type [Formula presented] in the nonstandard classes of random matrix theory of Altland and Zirnbauer. The relation between the noncolliding systems of the generalized meanders of Yor, which are associated with the even-dimensional Bessel processes, and the chiral random matrix theory is also clarified.

UR - http://www.scopus.com/inward/record.url?scp=85035196840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035196840&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.68.021112

DO - 10.1103/PhysRevE.68.021112

M3 - Article

AN - SCOPUS:85035196840

VL - 68

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 2

ER -