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 $n\to \infty$, 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.

Published at http://dx.doi.org/10.1214/105051607000000041 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• The Pearcey Process
• Differential Equations for Dyson Processes
• Vicious walk with a wall, noncolliding meanders, and chiral and Bogoliubov-deGennes random matrices

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• Noncolliding Brownian Motion and Determinantal Processes
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• Two Bessel Bridges Conditioned Never to Collide, Double Dirichlet Series, and Jacobi Theta Function
• Loop-free Markov chains as determinantal point processes