Constrained non-crossing Brownian motions, fermions and the Ferrari–Spohn distribution. Issue 3 (17th March 2021)

• Record Type: Journal Article
• Authors: Gautié, Tristan; Smith, Naftali R
• Is Part Of: Journal of statistical mechanics. Issue 3(2021)
• Publication Date: 2021-03-17
• Subjects: Brownian motion -- large deviations in non-equilibrium systems -- quantum gases -- stochastic particle dynamics; Statistical mechanics -- Periodicals; Mechanics -- Statistical methods -- Periodicals; 530.1305
• DOI: 10.1088/1742-5468/abe59c
• Languages: English
• ISSNs: 1742-5468

Abstract

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 × 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 ≫ 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.