The Annals of Probability

Airy processes with wanderers and new universality classes

Mark Adler, Patrik L. Ferrari, and Pierre van Moerbeke

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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.

Article information

Ann. Probab., Volume 38, Number 2 (2010), 714-769.

First available in Project Euclid: 9 March 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields 60G65 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 60G10: Stationary processes 35Q58

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


Adler, Mark; Ferrari, Patrik L.; van Moerbeke, Pierre. Airy processes with wanderers and new universality classes. Ann. Probab. 38 (2010), no. 2, 714--769. doi:10.1214/09-AOP493.

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