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 An−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 Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a semisimple Lie algebra, viewing it as a vector space with the Killing form. Since the Killing form is invariant under the adjoint, the motion induces a process in the Weyl chamber of the Lie algebra, 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 An−1 corresponding to un of n particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them.