Activated random walk exhibits self-organized criticality

TL;DR

This paper proves that the 1-dimensional activated random walk model self-organizes into a critical state with power-law avalanches, confirming a long-standing conjecture in self-organized criticality theory.

Contribution

It provides the first rigorous proof that the 1D activated random walk model exhibits self-organized criticality with quick mixing and power-law behavior.

Findings

Model mixes rapidly into a critical state

Power-law avalanches observed in the stationary state

Critical density matches the fixed-energy model's critical point

Abstract

To explain the ubiquity of power laws and fractals in nature, Bak, Tang, and Wiesenfeld formulated simple conditions for a system to self-organize into a critical state. Dickman, Mu\~noz, Vespignani, and Zapperi postulated that the self-organized critical state matches the critical state in corresponding fixed-energy models undergoing traditional phase transitions. Although the theory has been applied broadly over the past five decades, no mathematical model has been proven to exhibit the conjectured behavior. Indeed, the originally proposed abelian sandpile model displays nonuniversal behavior stemming from its slow mixing. Marking the first result of its kind, we prove that the 1-d activated random walk model mixes quickly into a stationary state with power-law avalanches and limiting critical density that equals the critical value for the fixed-energy version.