Boundary Asymptotics of Non-Intersecting Brownian Motions: Pearcey, Airy and a Transition

TL;DR

This paper analyzes the boundary behavior of non-intersecting Brownian motions, revealing three universal processes—Pearcey, Airy, and a new transition process—under mild assumptions, with applications to eigenvalue distributions.

Contribution

It introduces a new determinantal process describing the transition from Pearcey to Airy processes, expanding understanding of boundary asymptotics in non-intersecting Brownian motions.

Findings

Identification of three universal boundary processes

Derivation of a simple integral condition distinguishing cases

Application to eigenvalue distributions in random matrices

Abstract

We study $n$ non-intersecting Brownian motions, corresponding to the eigenvalues of an $n\times n$ 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 $n\to\infty$. Applications to largest eigenvalues of macro- and mesoscopic bulks and to random initial configurations are given.