Brownian Motions with One-Sided Collisions: The Stationary Case

Abstract

We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit $t\to \infty$. This leads to a new universal cross-over process.

55 pages, 10 figures

References in corpus (6)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Exact scaling functions for one-dimensional stationary KPZ growth
• Fluctuation properties of the TASEP with periodic initial configuration
• Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Two time distribution in Brownian directed percolation

Cited by in corpus (7)

• On Time Correlations for KPZ Growth in One Dimension
• Determinantal structures in the O'Connell-Yor directed random polymer model
• The hard-edge tacnode process for Brownian motion
• Finite GUE distribution with cut-off at a shock
• Fluctuations for stationary $q$-TASEP
• The half-space Airy stat process
• Large Deviation Principle for Interacting Brownian Motions