Universality of the Pearcey process

TL;DR

This paper demonstrates that at the bifurcation point in non-intersecting Brownian motions with two arbitrary targets, the paths exhibit Pearcey process fluctuations, establishing a universality result beyond symmetric cases.

Contribution

It extends the universality of the Pearcey process to asymmetric configurations and improves PDE results related to transition probabilities.

Findings

Pearcey process describes fluctuations at bifurcation points.

Universality holds for asymmetric target points.

Enhanced PDE results for transition probabilities.

Abstract

Consider non-intersecting Brownian motions on the line leaving from the origin and forced to two arbitrary points. Letting the number of Brownian particles tend to infinity, and upon rescaling, there is a point of bifurcation, where the support of the density of particles goes from one interval to two intervals. In this paper, we show that at that very point of bifurcation a cusp appears, near which the Brownian paths fluctuate like the Pearcey process. This is a universality result within this class of problems. Tracy and Widom obtained such a result in the symmetric case, when the two target points are symmetric with regard to the origin. This asymmetry enabled us to improve considerably a result concerning the non-linear partial differential equations governing the transition probabilities for the Pearcey process, obtained by Adler and van Moerbeke.