Nonintersecting Brownian excursions
Craig A. Tracy and Harold Widom
Source: Ann. Appl. Probab. Volume 17, Number 3
(2007), 953-979.
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.
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Keywords: Brownian excursions; nonintersecting paths; Karlin–McGregor; Fredholm determinants; Painlevé functions
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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1179839179
Digital Object Identifier: doi:10.1214/105051607000000041
Mathematical Reviews number (MathSciNet): MR2326237
Zentralblatt MATH identifier: 1124.60081
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