Two-Sided Infinite Systems of Competing Brownian Particles

TL;DR

This paper studies two-sided infinite systems of Brownian particles with rank-dependent dynamics, revealing unique stationary gap distributions and convergence behaviors not seen in finite or one-sided systems.

Contribution

It introduces the analysis of two-sided infinite systems, showing they can have diverse stationary gap distributions and different long-term behaviors.

Findings

Existence of one- or two-parameter families of stationary gap distributions.

The gap process can converge weakly to zero over time.

Distinct properties from finite and one-sided infinite systems.

Abstract

Two-sided infinite systems of Brownian particles with rank-dependent dynamics, indexed by all integers, exhibit different properties from their one-sided infinite counterparts, indexed by positive integers, and from finite systems. Consider the gap process, which is formed by spacings between adjacent particles. In stark contrast with finite and one-sided infinite systems, two-sided infinite systems can have one- or two-parameter family of stationary gap distributions, or the gap process weakly converging to zero as time goes to infinity.