The extreme statistics of some noncolliding Brownian processes

TL;DR

This paper studies noncolliding Brownian particle systems, establishing limit theorems for extremal particles, including eigenvalue distributions and process limits, with applications to random matrix theory and percolation models.

Contribution

It provides new limit theorems and formulas for the extremal behavior of noncolliding Brownian motions and related eigenvalue distributions, advancing understanding of these stochastic processes.

Findings

Scaling limit of the largest eigenvalue of Hermitian Brownian motion

Airy process limit for the largest eigenvalue of Dyson's Brownian motion

Fredholm determinant formula for the maximum of the top path among noncolliding Brownian bridges

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.