Activated random walk exhibits self-organized criticality

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.