Tobias Johnson: The density conjecture for activated random walk

Abstract

Physicists starting in the 1980s proposed that some systems display "self-organized criticality" to explain why real-life systems with no obvious phase transition exhibit self-similarity and power-law tails typical of statistical mechanics systems at criticality. The activated random walk model (ARW) has been proposed as a universal model for self-organized criticality. The crux of its behavior is the density conjecture: the system on a finite box with particles added in the middle and destroyed on the boundary naturally drives itself to the critical density of the infinite-lattice version of the model. We prove the density conjecture in one dimension, giving the first confirmation that the model demonstrates self-organized criticality in any setting. We also show that critical values for ARW in several other settings are identical, providing further evidence for the universality of ARW. Joint work with Chris Hoffman and Matt Junge.