Stationary Gap Distributions for Infinite Systems of Competing Brownian Particles

TL;DR

This paper characterizes a family of stationary gap distributions for infinite systems of competing Brownian particles, resolving a conjecture and extending results to more general drift configurations.

Contribution

It introduces a continuum family of stationary gap distributions for the Atlas model and generalizes to systems with arbitrary rank-based drifts.

Findings

Existence of infinitely many stationary gap distributions for the Atlas model.

Explicit form of the distributions as a family of product-of-exponentials.

Resolution of a conjecture by Pal and Pitman (2008).

Abstract

Consider the infinite Atlas model: a semi-infinite collection of particles driven by independent standard Brownian motions with zero drifts, except for the bottom-ranked particle which receives unit drift. We derive a continuum one-parameter family of product-of-exponentials stationary gap distributions, with exponentially growing density at infinity. This result shows that there are infinitely many stationary gap distributions for the Atlas model, and hence resolves a conjecture of Pal and Pitman (2008) in the negative. This result is further generalized for infinite systems of competing Brownian particles with generic rank-based drifts.