Non-fixation for Biased Activated Random Walks

TL;DR

This paper proves that biased Activated Random Walks on integer lattices do not fixate under certain conditions, using a new non-fixation criterion and a pathwise construction of the process.

Contribution

It introduces a novel non-fixation criterion and a pathwise construction method for biased Activated Random Walks, advancing understanding of their long-term behavior.

Findings

Non-fixation for any positive density with small sleep rate.

Non-fixation for high density close to 1 regardless of sleep rate.

Development of a new criterion for non-fixation in activated random walks.

Abstract

We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.