The extreme statistics of some noncolliding Brownian processes

Abstract

We consider certain noncolliding interacting particle systems driven by Brownian noise. A key example is drifted Brownian motions conditioned not to intersect and related models of eigenvalues of Hermitian random matrices. We establish limit theorems for the extremal particle. We find: (i) the scaling limit of the largest eigenvalue of Brownian motion over Hermitian, positive-definite matrices, (ii) Airy process limit for the largest eigenvalue of Dyson's Brownian motion for GUE started from generic initial conditions, and (iii) a Fredholm determinant formula for the maximum of the top path among noncolliding Brownian bridges and, as a byproduct, a new formula for the law of largest eigenvalue in a particular Laguerre Orthogonal Ensemble as well as for a related point-to-line last passage percolation model.