Noncolliding processes, matrix-valued processes and determinantal processes

TL;DR

This paper explores noncolliding diffusion processes, their connection to random matrix eigenvalues, and their determinantal structure, including asymptotic behavior as the number of particles grows large.

Contribution

It introduces generalized noncolliding diffusion processes, including inhomogeneous cases, and analyzes their determinantal properties and asymptotic behaviors in relation to random matrix theory.

Findings

Noncolliding processes are determinantal point processes.

The multi-time correlation functions are represented by determinants.

Asymptotic analysis links to eigenvalue distributions in large matrices.

Abstract

A noncolliding diffusion process is a conditional process of $N$ independent one-dimensional diffusion processes such that the particles never collide with each other. This process realizes an interacting particle system with long-ranged strong repulsive forces acting between any pair of particles. When the individual diffusion process is a one-dimensional Brownian motion, the noncolliding process is equivalent in distribution with the eigenvalue process of an $N \times N$ Hermitian-matrix-valued process, which we call Dyson's model. For any deterministic initial configuration of $N$ particles, distribution of particle positions of the noncolliding Brownian motion on the real line at any fixed time $t >0$ is a determinantal point process. We can prove that the process is determinantal in the sense that the multi-time correlation function for any chosen series of times, which determines joint distributions at these times, is also represented by a determinant. We study the asymptotic behavior of the system, when the number of Brownian motions $N$ in the system tends to infinity. This problem is concerned with the random matrix theory on the asymptotics of eigenvalue distributions, when the matrix size becomes infinity. In the present paper, we introduce a variety of noncolliding diffusion processes by generalizing the noncolliding Brownian motion, some of which are temporally inhomogeneous. We report the results of our research project to construct and study finite and infinite particle systems with long-ranged strong interactions realized by noncolliding processes.