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June 2007 Nonintersecting Brownian excursions
Craig A. Tracy, Harold Widom
Ann. Appl. Probab. 17(3): 953-979 (June 2007). DOI: 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 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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Craig A. Tracy. Harold Widom. "Nonintersecting Brownian excursions." Ann. Appl. Probab. 17 (3) 953 - 979, June 2007.


Published: June 2007
First available in Project Euclid: 22 May 2007

zbMATH: 1124.60081
MathSciNet: MR2326237
Digital Object Identifier: 10.1214/105051607000000041

Primary: 33E17 , 60J65 , 60K35

Keywords: Brownian excursions , Fredholm determinants , Karlin–McGregor , nonintersecting paths , Painlevé functions

Rights: Copyright © 2007 Institute of Mathematical Statistics


Vol.17 • No. 3 • June 2007
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