Local Density of Activated Random Walk on $\mathbb{Z}$

TL;DR

This work proves a local density theorem for the one-dimensional activated random walk started from a large point source: in a macroscopic window around the origin, the probability that a site hosts a sleeping particle after stabilization converges to the critical density $\rho_c$, uniformly in the window. The authors develop a novel two-phase IDLA-based framework, coupling two IDLA clusters and analyzing block densities via a driven-dissipative ARW on finite intervals, to show near-constancy of local sleeping probabilities and convergence of block densities to $\rho_c$. From these ingredients, they deduce that any subsequential local limit of the point-source laws, if it exists, is shift-invariant with density $\rho_c$, thereby confirming part of the Levine–Silvestri universality picture in 1D. The paper also outlines directions for extending the approach to higher dimensions and to multi-site statistics, strengthening the link between local structure and global criticality in ARW.

Abstract

We consider one-dimensional activated random walk (ARW) on $\mathbb{Z}$ started from a `point source' initial condition, with many particles at the origin and no other particles. We prove that, uniformly throughout a macroscopic window around the source, the probability that a site contains a sleeping particle after the configuration is stabilized is approximately the critical density. This represents a first step towards understanding the local structure of the critical stationary measure for ARW.

Theorems & Definitions (18)

• Theorem 1.1
• Proposition 1.2
• Lemma 2.1: Abelian Property rolla2020activated
• Theorem 3.1: Theorem 1 of Mittelstaedt19
• Lemma 3.2
• proof
• Theorem 3.3: Theorem 1.1 of Ozdemir22
• Lemma 3.4
• proof
• Lemma 3.5