A new proof of superadditivity and of the density conjecture for Activated Random Walks on the line

TL;DR

This paper provides a new, more elementary proof of superadditivity and the density conjecture for Activated Random Walks on the line, potentially enabling extensions beyond one dimension.

Contribution

It introduces a novel superadditivity property for Activated Random Walks, offering an alternative proof to existing conjectures without using percolation comparison.

Findings

Established a new superadditivity property for Activated Random Walks.

Provided a different proof of the density and related conjectures.

Suggested potential for extending results beyond one dimension.

Abstract

In two recent works, Hoffman, Johnson and Junge proved the density conjecture, the hockey stick conjecture and the ball conjecture for Activated Random Walks in dimension one, showing an equality between several different definitions of the critical density of the model. This establishes a kind of self-organized criticality, which was originally predicted for the Abelian Sandpile Model. Their proof uses a comparison with a percolation process, which exhibits superadditivity. We present here a different proof of these conjectures, based on a new superadditivity property that we establish directly for Activated Random Walks, without relying on a percolation process. This more elementary approach yields less precise bounds than the percolation technology developed by Hoffman, Johnson and Junge, but it might open new perspectives to go beyond the one-dimensional setting.