Law of large numbers for activated random walk on villages

TL;DR

This paper studies a generalized activated random walk model on complex graph structures called villages, proving a law of large numbers for the stable configuration as the village size grows, under subcritical conditions.

Contribution

It introduces the village model of ARW (VARW) on arbitrary graphs and establishes a law of large numbers for the stable state in the large village limit.

Findings

Proves LLN for stable configurations in VARW as village size increases.

Characterizes the limit via a system of nonlinear equations.

Ensures almost sure stabilization under subcritical initial conditions.

Abstract

We consider activated random walk (ARW), an interacting particle system and prototypical model of self-organized criticality in a setting which combines mean-field behavior with the geometry of an arbitrary graph, which we call the village model of ARW, or VARW for short. VARW is obtained from a fixed graph by replacing each vertex with a 'village' that consists of n replicas of that vertex. We focus on VARW where particles walk according to a strictly sub-stochastic transition kernel on a finite underlying graph, so mass is sometimes lost (which guarantees that the system eventually stabilizes almost surely). Under a subcriticality assumption on the initial state we prove a law of large numbers as n goes to infinity for the resulting stable configuration of particles and the odometer of the process, to a limit which is uniquely characterized by a system of non-linear equations.