Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution

TL;DR

This paper extends the Ferrari-Spohn model to multiple non-crossing Brownian bridges, revealing their joint distribution relates to fermions in a linear potential, and explores large N behavior with a semicircular wall.

Contribution

It generalizes the Ferrari-Spohn model to N non-crossing Brownian bridges and connects their distribution to fermionic systems, providing insights into large N asymptotics.

Findings

Joint distribution expressed as a determinant involving Airy functions.

Distribution matches that of noninteracting fermions in a linear potential.

Large N behavior analyzed for a semicircular wall trajectory.

Abstract

A conditioned stochastic process can display a very different behavior from the unconditioned process. In particular, a conditioned process can exhibit non-Gaussian fluctuations even if the unconditioned process is Gaussian. In this work, we revisit the Ferrari-Spohn model of a Brownian bridge conditioned to avoid a moving wall, which pushes the system into a large-deviation regime. We extend this model to an arbitrary number $N$ of non-crossing Brownian bridges. We obtain the joint distribution of the distances of the Brownian particles from the wall at an intermediate time in the form of the determinant of an $N\times N$ matrix whose entries are given in terms of the Airy function. We show that this distribution coincides with that of the positions of $N$ spinless noninteracting fermions trapped by a linear potential with a hard wall. We then explore the $N \gg 1$ behavior of the system. For simplicity we focus on the case where the wall's position is given by a semicircle as a function of time, but we expect our results to be valid for any concave wall function.