Equilibrium Fluctuation of the Atlas Model

Abstract

We study the fluctuation of the Atlas model, where a unit drift is assigned to the lowest ranked particle among a semi-infinite (${\mathbb{Z}}_{+}$-indexed) system of otherwise independent Brownian particles, initiated according to a Poisson point process on ${\mathbb{R}}_{+}$. In this context, we show that the joint law of ranked particles, after being centered and scaled by $t^{-1/4}$, converges as $t\to \infty$ to the Gaussian field corresponding to the solution of the additive stochastic heat equation on ${\mathbb{R}}_{+}$ with Neumann boundary condition at zero. This allows us to express the asymptotic fluctuation of the lowest ranked particle in terms of a $\frac{1}{4}$-fractional Brownian motion. In particular, we prove a conjecture of Pal and Pitman (2008) about the asymptotic Gaussian fluctuation of the ranked particles.

arXiv comment28 pages