Dyson's non-intersecting Brownian motions with a few outliers

TL;DR

This paper investigates the fluctuations of non-intersecting Brownian motions with a few outliers, revealing a new r-Airy process with extended kernel and PDE, generalizing the classical Airy process.

Contribution

It introduces the r-Airy process as a new fluctuation process for Dyson Brownian motions with outliers, extending the Airy process and kernel.

Findings

The r-Airy process generalizes the Airy process for outliers.

The transition probability is expressed via a Fredholm determinant of an extended kernel.

The process satisfies a non-linear PDE in space and time.

Abstract

Consider n non-intersecting particles on the real line (Dyson Brownian motions), all starting from the origin at time=0, and forced to return to x=0 at time=1. For large n, the average mean density of particles has its support, for each 0<t<1, within the interior of an ellipse. The Airy process is defined as the motion of these non-intersecting Brownian motions for large n, but viewed from an arbitrary point on the ellipse with an appropriate space-time rescaling. Assume now a finite number r of these particles are forced to a different target point. Does it affect the Brownian fluctuations along the ellipse for large n? In this paper, we show that no new process appears as long as one considers points on the ellipse, for which the t-coordinate is smaller than the t-coordinate of the point of tangency of the tangent to the curve passing through the target point. At this point of tangency the fluctuations obey a new statistics: the Airy process with r outliers (in short: {\bf r-Airy process}). The log of the transition probability of this new process is given by the Fredholm determinant of a new kernel (extending the Airy kernel) and it satisfies a non-linear PDE in x and the time.