Noncolliding Brownian motion and determinantal processes

Makoto Katori, Hideki Tanemura

Research output: Contribution to journalArticle

38 Citations (Scopus)

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 languageEnglish
Pages (from-to)1233-1277
Number of pages45
JournalJournal of Statistical Physics
Volume129
Issue number5-6
DOIs
Publication statusPublished - 2007 Oct 1
Externally publishedYes

Fingerprint

Brownian motion
matrices
determinants
kernel
Absorbing
eigenvalues
Ensemble
harmonic functions
H-transform
Vandermonde determinant
Spectral Projection
Eigenvalue Distribution
Transition Density
continuity
Scaling Limit
transition probabilities
Particle System
Hermitian matrix
Harmonic Functions
Transition Probability

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

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

In: Journal of Statistical Physics, Vol. 129, No. 5-6, 01.10.2007, p. 1233-1277.

Research output: Contribution to journalArticle

@article{f0fe94ad601b4b9eb35a1fd68a98285d,
title = "Noncolliding Brownian motion and determinantal processes",
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.",
keywords = "Determinantal processes, Karlin-McGregor formula, Matrix-kernels, Multitime correlation functions, Noncolliding Brownian motion, Random matrix theory, Spectral projections",
author = "Makoto Katori and Hideki Tanemura",
year = "2007",
month = "10",
day = "1",
doi = "10.1007/s10955-007-9421-y",
language = "English",
volume = "129",
pages = "1233--1277",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
number = "5-6",

}

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 -