Dyson's Nonintersecting Brownian Motions with a Few Outliers

Consider n nonintersecting Brownian particles on R (Dyson Brownian motions), all starting from the origin at time t = 0 and forced to return to X = 0 at time t = 1. For large n, the average mean density of particles has its Support, for each 0 < t < 1, on the interval +/-root 2nt (1 - t). The Airy process A(tau) is defined as the Motion of these nonintersecting Brownian motions for large n but viewed from the curve C : y = root 2nt (1 - t) with an appropriate space-time rescaling. Assume now a finite number r of these particles ire forced to a different target point, say a = rho 0 root n/2 > 0. Does it affect the Brownian fluctuations along the Curve C for large n? In this paper, we show that no new process appears as long as one considers points (y, t) is an element of C such that 0 < t < (1 + rho(2)(0))(-1), which is the t-coordinate of the point of tangency of the tangent to the curve passing through (rho(0) root n/2, 1). At this point the fluctuations obey a new statistics, which we call the Air), process with r outhers A((r)) (tau) (in short, r-Airy process). The log of the probability that at time r the cloud does not exceed x is given by the Fredholm determinant of a new kernel (extending the Airy kernel), and it satisfies a nonlinear PDE in x and tau, from which the asymptotic behavior of the process can be deduced for tau -> -infinity. This kernel is closely related to one found by Baik, Ben Arous, and Peche in the context of multivariate statistics. (C) 2008 Wiley Periodicals, Inc.