Macroscopic flow out of a segment for Activated Random Walks in dimension 1

TL;DR

This paper proves that in one-dimensional Activated Random Walks, a positive fraction of particles exit a segment in the supercritical phase, confirming a key condition for the active phase and addressing conjectures about the phase transition.

Contribution

It establishes the necessity of a known sufficient condition for activity in the supercritical phase of one-dimensional Activated Random Walks.

Findings

Positive fraction of particles leave the segment in supercritical case

Condition for being in the active phase is also necessary

Addresses conjectures on phase transition of the model

Abstract

Activated Random Walk is a system of interacting particles which presents a phase transition and a conjectured phenomenon of self-organized criticality. In this note, we prove that, in dimension 1, in the supercritical case, when a segment is stabilized with particles being killed when they jump out of the segment, a positive fraction of the particles leaves the segment with positive probability. This was already known to be a sufficient condition for being in the active phase of the model, and the result of this paper is that this condition is also necessary, except maybe precisely at the critical point. This result can also be seen as a partial answer to some of the many conjectures which connect the different points of view on the phase transition of the model.