Noncolliding Brownian motion and determinantal processes

Makoto Katori, Hideki Tanemura

Research output: Contribution to journalArticle

40 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

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

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