The density conjecture for activated random walk

TL;DR

This paper proves the long-standing density conjecture for activated random walk in one dimension, confirming its role as a universal model of self-organized criticality by establishing the convergence of particle density to the critical value.

Contribution

It provides the first rigorous proof of the density conjecture for activated random walk and shows the universality of the model across different variants.

Findings

Proved the density conjecture for one-dimensional activated random walk.

Established that different natural versions share the same critical density.

Confirmed activated random walk as a universal self-organized criticality model.

Abstract

Bak, Tang, and Wiesenfeld developed their theory of self-organized criticality in the late 1980s to explain why many real-life processes exhibit signs of critical behavior despite the absence of a tuning parameter. A decade later, Dickman, Mu\~noz, Vespignani, and Zapperi explained self-organized criticality as an external force pushing a hidden parameter toward the critical value of a traditional absorbing-state phase transition. As evidence, they observed empirically that for various sandpile models, the particle density in a finite box under driven-dissipative dynamics converges to the critical density of an infinite-volume version of the model. We give the first proof of this well-known density conjecture in any setting by establishing it for activated random walk in one dimension. We prove that two other natural versions of the model have the same critical value, further establishing activated random walk as a universal model of self-organized criticality.