Noncolliding Brownian Motion and Determinantal Processes

TL;DR

This paper explores noncolliding Brownian motions, showing they form determinantal processes with explicit kernels, and derives infinite particle limits with applications in random matrix theory and spectral analysis.

Contribution

It demonstrates that noncolliding Brownian motions are determinantal processes with explicit correlation kernels, extending understanding of their structure and scaling limits.

Findings

Noncolliding Brownian motions form determinantal processes.

Explicit matrix-kernels are derived using spectral projections.

Infinite determinantal processes are obtained through scaling limits.

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.