Scaling limit for Brownian motions with one-sided collisions

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 Schütz-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.

arXiv commentPublished at http://dx.doi.org/10.1214/14-AAP1025 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• The one-dimensional KPZ equation and its universality class
• The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions
• Reflected Brownian motions in the KPZ universality class
• One-sided reflected Brownian motions and the KPZ fixed point
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Large Deviations For Sticky Brownian Motions
• Height fluctuations for the stationary KPZ equation
• Brownian Motions with One-Sided Collisions: The Stationary Case
• KPZ universality class and the anchored Toom interface
• Exact solution of interacting particle systems related to random matrices
• Large Deviation Principle for Interacting Brownian Motions
• Triple and Simultaneous Collisions of Competing Brownian Particles