Asymptotics of non-intersecting Brownian motions and a 4×4 Riemann–Hilbert problem

AbstractWe consider n one-dimensional Brownian motions, such that n/2 Brownian motions start at time t=0 in the starting point a and end at time t=1 in the endpoint b and the other n/2 Brownian motions start at time t=0 at the point -a and end at time t=1 in the point -b, conditioned that the n Brownian paths do not intersect in the whole time interval (0,1). The correlation functions of the positions of the non-intersecting Brownian motions have a determinantal form with a kernel that is expressed in terms of multiple Hermite polynomials of mixed type. We analyze this kernel in the large n limit for the case ab