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

Tristan Gautié, Naftali R. Smith

Research output: Contribution to journalArticlepeer-review


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.

Original languageAmerican English
Article number033212
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number3
StatePublished - 1 Mar 2021
Externally publishedYes


  • Brownian motion
  • large deviations in non-equilibrium systems
  • quantum gases
  • stochastic particle dynamics

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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