Equilibrium Fluctuation of the Atlas Model

TL;DR

This paper analyzes the long-term fluctuation behavior of the Atlas model, showing convergence to a Gaussian field and confirming a conjecture about the Gaussian fluctuations of ranked particles.

Contribution

It establishes the asymptotic Gaussian fluctuation of the Atlas model's ranked particles, connecting it to the additive stochastic heat equation with Neumann boundary conditions.

Findings

Joint law converges to Gaussian field after scaling

Lowest ranked particle fluctuation described by fractional Brownian motion

Confirms Pal and Pitman's conjecture on Gaussian fluctuation

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.