Constrained non-crossing Brownian motions, fermions and the Ferrari–Spohn distribution – Archive ouverte HAL

Tristan Gautié 1 Naftali R. Smith 2, 1

Tristan Gautié, Naftali R. Smith. Constrained non-crossing Brownian motions, fermions and the Ferrari–Spohn distribution. J.Stat.Mech., 2021, 2103, pp.033212. ⟨10.1088/1742-5468/abe59c⟩. ⟨hal-03186169⟩

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.

  • 1. Champs Aléatoires et Systèmes hors d'Équilibre
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques