The Annals of Probability
- Ann. Probab.
- Volume 38, Number 2 (2010), 714-769.
Airy processes with wanderers and new universality classes
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 , 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.
Ann. Probab., Volume 38, Number 2 (2010), 714-769.
First available in Project Euclid: 9 March 2010
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.aop/1268143531