1997/08/20 by David J. Grabiner, Grabiner, David J. · 4 citations
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Representation Theory (math.RT) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.math/9708207
openalex publication_date 1997/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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 An-1 corresponding to \mathfrak s\mathfrak un of n particles moving in a diffusion with a repelling force between two particles proportional to the inverse of the distance between them.