# Stationary Distributions of the Atlas Model

## TL;DR

This paper analyzes the Atlas model with Brownian particles, establishing explicit invariant distributions and gap distributions, and provides bounds on the fluctuation of the lowest-ranked particle.

## Contribution

It derives explicit invariant and gap distributions for the Atlas model and proves fluctuation bounds for the lowest-ranked particle.

## Key findings

- Invariant distribution $
u_a$ explicitly characterized
- Product-of-exponential gap distribution $ ext{pi}_a$ confirmed
- Bound on the fluctuation of the lowest particle established

## Abstract

In this article we study the Atlas model, which constitutes of Brownian particles on $ \mathbb{R} $, independent except that the Atlas (i.e., lowest ranked) particle $ X_{(1)}(t) $ receive drift $ \gamma dt $, $ \gamma\in\mathbb{R} $. For any fixed shape parameter $ a>2\gamma_- $, we show that, up to a shift $ \frac{a}{2}t $, the entire particle system has an invariant distribution $ \nu_a $, written in terms an explicit Radon-Nikodym derivative with respect to the Poisson point process of density $ a e^{a\xi} d\xi $. We further show that $ \nu_a $ indeed has the product-of-exponential gap distribution $ \pi_a $ derived in Sarantsev and Tsai (2016). As a simple application, we establish a bound on the fluctuation of the Atlas particle $ X_{(1)}(t) $ uniformly in $ t $, with the gaps initiated from $ \pi_a $ and $ X_{(1)}(0)=0 $.