Critical Behavior of Non-Intersecting Brownian Motions

TL;DR

This paper investigates the microscopic correlations of non-intersecting Brownian motions near a critical point where the density vanishes, revealing connections to the Pearcey process and Airy line ensemble, with surprising interior spectrum behavior.

Contribution

It demonstrates the emergence of Pearcey and Airy processes in the interior of the spectrum, explaining the unexpected appearance of the Airy line ensemble away from edges.

Findings

Microscopic correlations described by Pearcey or Airy processes.

Interior spectrum exhibits mesoscopic gaps leading to Airy process appearance.

Identification of a path following the Airy$_2$ process.

Abstract

We study $n$ non-intersecting Brownian motions corresponding to initial configurations which have a vanishing density in the large $n$ limit at an interior point of the support. It is understood that the point of vanishing can propagate up to a critical time, and we investigate the nature of the microscopic space-time correlations near the critical point and critical time. We show that they are described either by the Pearcey process or by the Airy line ensemble, depending on whether a simple integral related to the initial configuration vanishes or not. Since the Airy line ensemble typically arises near edge points of the macroscopic density, its appearance in the interior of the spectrum is surprising. We explain this phenomenon by showing that, even though there is no gap of macroscopic size near the critical point, there is with high probability a gap of mesoscopic size. Moreover, we identify a path which follows the Airy$_2$ process.