Nonintersecting Brownian excursions

Details Nonintersecting Brownian excursions Nonintersecting Brownian excursions

2006-07-13

arxiv.org/abs/math/0607321v3

Annals of Applied Probability 2007, Vol. 17, No. 3, 953-979

Mathematics

Probability

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

Scientific paper

10.1214/105051607000000041

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 Painlev\'{e} 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.

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