Nonintersecting Brownian excursions

Abstract

We consider the process of n Brownian excursions conditioned to be nonintersecting. We show the distribution functions for the top curve and the bottom curve are equal to Fredholm determinants whose kernel we give explicitly. In the simplest case, these determinants are expressible in terms of Painlevé V functions. We prove that as n→∞, the distributional limit of the bottom curve is the Bessel process with parameter 1/2. (This is the Bessel process associated with Dyson’s Brownian motion.) We apply these results to study the expected area under the bottom and top curves.