The hockey-stick conjecture for activated random walk

TL;DR

This paper rigorously confirms that the activated random walk model self-organizes to a critical density, providing the first proof of a sandpile model exhibiting self-organized criticality as originally theorized.

Contribution

It proves a conjecture that the activated random walk model on an interval self-organizes to a critical state, confirming a key aspect of self-organized criticality in sandpile models.

Findings

The model reaches and maintains a critical density.

First rigorous proof of self-organized criticality in a sandpile model.

Supports the original vision of Bak, Tang, and Wiesenfeld.

Abstract

We prove a conjecture of Levine and Silvestri that the driven-dissipative activated random walk model on an interval drives itself directly to and then sustains a critical density. This marks the first rigorous confirmation of a sandpile model behaving as in Bak, Tang, and Wiesenfeld's original vision of self-organized criticality.