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

The paper investigates a system of one‑dimensional Brownian particles conditioned never to collide, establishing deep connections between random matrix theory, stochastic processes, and determinantal point fields. Two equivalent constructions are presented. First, Dyson’s Brownian motion model arises as the eigenvalue process of an N×N Hermitian matrix‑valued diffusion (the Gaussian Unitary Ensemble, GUE). The eigenvalues satisfy the stochastic differential equations
 dX_j(t)=dB_j(t)+∑{k≠j} dt/(X_j(t)−X_k(t)), j=1,…,N,
which generalize the three‑dimensional Bessel process. Second, the same non‑colliding system is obtained by imposing absorbing walls on the Weyl chamber W_N={x∈ℝ^N:x_1<⋯<x_N} for independent Brownian motions and performing an h‑transform with the Vandermonde harmonic function h_N(x)=∏{j<k}(x_k−x_j). The Karlin‑McGregor formula yields the transition density of the absorbing process as a determinant of heat kernels, and the h‑transform produces the non‑colliding transition density
 p_N(t,y|x)=h_N(y)/h_N(x)·det