Noncolliding Brownian Motion and Determinantal Processes

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.

💡 Research Summary

This paper investigates a system of one‑dimensional Brownian particles conditioned never to intersect, a model commonly called non‑colliding Brownian motion (NBM). The authors present two mathematically equivalent constructions of NBM. The first is Dyson’s Brownian‑motion model: a Hermitian matrix‑valued diffusion process whose entries evolve as independent complex Brownian motions. The eigenvalues of this matrix follow a stochastic dynamics that never collide, and when the initial eigenvalue distribution is taken from the Gaussian Unitary Ensemble (GUE) the process remains in the GUE class for all times. The second construction uses an absorbing Brownian motion confined to the Weyl chamber (\mathcal{W}N={x_1<\dots<x_N}). By performing an (h)-transform with the harmonic function
(h(\mathbf{x})=\prod{1\le i<j\le N}(x_j-x_i)) (the Vandermonde determinant), the absorbing process is turned into a non‑absorbing one that automatically avoids collisions. This (h)-transform is precisely equivalent to Dyson’s eigenvalue dynamics, establishing a one‑to‑one correspondence between the two viewpoints.

A central technical tool is the Karlin–McGregor formula, which expresses the transition density of (N) independent one‑dimensional Brownian motions conditioned never to intersect as a determinant of the single‑particle heat kernel:
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