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

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.

The Airy2 process and the 3D Ising model

“…For the special case of the one-point distribution, the original work [15] on Brownian motion conditioned to stay above a large circle, was extended in a physical level of rigor in [17] to several walks and a physical derivation of the one-point case is provided.…”

Matrix Kesten recursion, inverse-Wishart ensemble and fermions in a Morse potential

“…We now apply these results to our Kesten matrix problem and use the parameter values g = N − m σ 2 and e −x0 = σ 2 g given in (54). In that case the effective Fermi energy (73) becomes…”