Scaling limit for Brownian motions with one-sided collisions

TL;DR

This paper studies Brownian particles with one-sided reflection, deriving formulas for finite systems, and shows that in large time limits, their fluctuations follow the Airy$_1$ process, revealing universal behavior in such stochastic systems.

Contribution

It provides a Sch"{u}tz-type formula for finite systems and characterizes the large-time fluctuations as the Airy$_1$ process for an infinite periodic system.

Findings

Derived transition probability formula for finite particles

Expressed joint distributions as Fredholm determinants

Fluctuations converge to the Airy$_1$ process in large time limit

Abstract

We consider Brownian motions with one-sided collisions, meaning that each particle is reflected at its right neighbour. For a finite number of particles a Sch\"{u}tz-type formula is derived for the transition probability. We investigate an infinite system with periodic initial configuration, that is, particles are located at the integer lattice at time zero. The joint distribution of the positions of a finite subset of particles is expressed as a Fredholm determinant with a kernel defining a signed determinantal point process. In the appropriate large time scaling limit, the fluctuations in the particle positions are described by the Airy$_1$ process.