A new proof of superadditivity and of the density conjecture for Activated Random Walks on the line (2502.02579v1)

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 1, showing an equality between several different definitions of the critical density of the model. This establishes a kind of self-organized criticality, that was originally predicted for the Abelian Sandpile Model. The proof of Hoffman, Johnson and Junge uses a comparison with a percolation process, which exhibits a superadditivity property. In the present note, we revisit their argument by providing a new proof of superadditivity directly for Activated Random Walks, without relying on a percolation process. The proof relies on a simple comparison between the stabilization of two neighbouring segments and that of their union. We then explain how this superaddivity property implies the three mentioned conjectures. Yet, so far it does not seem that this approach yields as much information as does the percolation technology developed by Hoffman, Johnson and Junge, which yields an exponential concentration bound on the stationary density, whereas the superadditivity property alone only ensures an exponential bound on the lower tail.