Avalanches in Critical Activated Random Walks

TL;DR

This paper studies the behavior of activated random walks on the integer lattice at critical density, revealing that the flow of particles through the origin converges to a self-similar pure-jump process, with explicit description.

Contribution

It introduces a new scaling limit for particle flow in asymmetric activated random walks at critical density, explicitly characterizing the limiting process.

Findings

Flow of particles converges to a self-similar pure-jump process.

Explicit description of the limiting process.

Demonstrates different time scales for particle release and dissipation.

Abstract

We consider Activated Random Walks on $\Z$ with totally asymmetric jumps and critical particle density, with different time scales for the progressive release of particles and the dissipation dynamics. We show that the cumulative flow of particles through the origin rescales to a pure-jump self-similar process which we describe explicitly.