Probability and Stochastic Analysis - University of Bonn

Abstract of "Airy processes with wanderers and new universality classes"

with Mark Adler and Pierre Van Moerbeke.

Consider n+m non-intersecting 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 a_i to m points b_i. First we keep m fixed and we scale a_i,b_i appropriately with n. In the large-n limit we obtain a new Airy process with wanderers, in the neighborhood of (2n)1/2, 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 we conjecture to be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.

Postscript file: [PS]

PDF file: [PDF]

arXiv: 0811.1863