Boundary asymptotics of non-intersecting Brownian motions: Pearcey, Airy and a transition

Abstract

We study n non-intersecting Brownian motions, corresponding to the eigenvalues of an n × n Hermitian Brownian motion. At the boundary of their limit shape we find that only three universal processes can arise: the Pearcey process close to merging points, the Airy line ensemble at edges and a novel determinantal process describing the transition from the Pearcey process to the Airy line ensemble. The three cases are distinguished by a remarkably simple integral condition. Our results hold under very mild assumptions, in particular we do not require any kind of convergence of the initial configuration as n→∞ . Applications to largest eigenvalues of macro- and mesoscopic bulks and to random initial configurations are given.