Brownian Motion in a Weyl Chamber, Non-Colliding Particles, and Random Matrices

Abstract

Let $n$ particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is $A_{n-1}$, the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time $t$ without having collided by time $t$. We show that the probability that there will be no collision up to time $t$ is asymptotic to a constant multiple of $t^{-n(n-1)/4}$ as $t$ goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group $B_n$ gives a model of $n$ independent particles with a wall at $x=0$. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point in the Weyl chamber, giving a Brownian motion conditioned never to exit the chamber. If there are $m$ roots in $n$ dimensions, this shows that the radial part of the conditioned process is the same as the $n+2m$-dimensional Bessel process. The conditioned process also gives physical models, generalizing Dyson's model for $A_{n-1}$ corresponding to $\mathfrak{s}{\mathfrak{u}}_n$ of $n$ particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them.