Activated Random Walks on $\mathbb{Z}$ with Critical Particle Density

TL;DR

This paper proves that the Activated Random Walks model on the integer lattice maintains its critical density even when starting with only one active particle, reinforcing its universality in self-organized criticality.

Contribution

It extends previous results by showing the critical density holds with a single active particle initially, relaxing earlier assumptions and strengthening the model's universality.

Findings

Critical density is preserved with a single active particle.

Supports the universality of the ARW model in 1D.

Demonstrates robustness of the critical behavior.

Abstract

The Activated Random Walk (ARW) model is a promising candidate for demonstrating self-organized criticality due to its potential for universality. Recent studies have shown that the ARW model exhibits a well-defined critical density in one dimension, supporting its universality. In this paper, we extend these results by demonstrating that the ARW model on $\mathbb{Z}$, with a single initially active particle and all other particles sleeping, maintains the same critical density. Our findings relax the previous assumption that required all particles to be initially active. This provides further evidence of the ARW model's robustness and universality in depicting self-organized criticality.