Local Density of Activated Random Walk on $\mathbb{Z}$

TL;DR

This paper investigates the local behavior of activated random walk on the integer lattice, showing that near the source, the probability of sleeping particles aligns with the critical density, advancing understanding of the model's critical measure.

Contribution

It provides the first analysis of local particle configurations near the source in one-dimensional ARW at criticality.

Findings

Probability of sleeping particles near the source approximates the critical density.

Uniform behavior observed throughout a macroscopic window around the source.

Advances understanding of the local structure of the critical stationary measure.

Abstract

We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW.