### Abstract

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

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

Pages (from-to) | 1233-1277 |

Number of pages | 45 |

Journal | Journal of Statistical Physics |

Volume | 129 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - 2007 Oct 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Determinantal processes
- Karlin-McGregor formula
- Matrix-kernels
- Multitime correlation functions
- Noncolliding Brownian motion
- Random matrix theory
- Spectral projections

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*129*(5-6), 1233-1277. https://doi.org/10.1007/s10955-007-9421-y

**Noncolliding Brownian motion and determinantal processes.** / Katori, Makoto; Tanemura, Hideki.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 129, no. 5-6, pp. 1233-1277. https://doi.org/10.1007/s10955-007-9421-y

}

TY - JOUR

T1 - Noncolliding Brownian motion and determinantal processes

AU - Katori, Makoto

AU - Tanemura, Hideki

PY - 2007/10/1

Y1 - 2007/10/1

N2 - A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

AB - A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

KW - Determinantal processes

KW - Karlin-McGregor formula

KW - Matrix-kernels

KW - Multitime correlation functions

KW - Noncolliding Brownian motion

KW - Random matrix theory

KW - Spectral projections

UR - http://www.scopus.com/inward/record.url?scp=36448991470&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36448991470&partnerID=8YFLogxK

U2 - 10.1007/s10955-007-9421-y

DO - 10.1007/s10955-007-9421-y

M3 - Article

VL - 129

SP - 1233

EP - 1277

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -