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March 2010 Airy processes with wanderers and new universality classes
Mark Adler, Patrik L. Ferrari, Pierre van Moerbeke
Ann. Probab. 38(2): 714-769 (March 2010). DOI: 10.1214/09-AOP493


Consider n+m nonintersecting Brownian bridges, with n of them leaving from 0 at time t=−1 and returning to 0 at time t=1, while the m remaining ones (wanderers) go from m points ai to m points bi. First, we keep m fixed and we scale ai, bi appropriately with n. In the large-n limit, we obtain a new Airy process with wanderers, in the neighborhood of $\sqrt{2n}$, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation.

Letting the number m of wanderers tend to infinity as well, leads to two Pearcey processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Upon tuning the starting and target points, one can let the two tips of the cusps grow very close; this leads to a new process, which might be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.


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Mark Adler. Patrik L. Ferrari. Pierre van Moerbeke. "Airy processes with wanderers and new universality classes." Ann. Probab. 38 (2) 714 - 769, March 2010.


Published: March 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1200.60069
MathSciNet: MR2642890
Digital Object Identifier: 10.1214/09-AOP493

Primary: 35Q53 , 60G60 , 60G65
Secondary: 35Q58 , 60G10

Keywords: Airy process , coupled random matrices , Dyson’s Brownian motion , extended kernels , Pearcey process , quintic kernel , random Hermitian ensembles

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 2 • March 2010
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