Active Phase for Activated Random Walks on the Lattice in all Dimensions
Abstract
We show that the critical density of the Activated Random Walk model on $\mathbb{Z}^d$ is strictly less than one when the sleep rate $\lambda$ is small enough, and tends to $0$ when $\lambda\to 0$, in any dimension $d\geqslant 1$. As far as we know, the result is new for $d=2$. We prove this by showing that, for high enough density and small enough sleep rate, the stabilization time of the model on the $d$dimensional torus is exponentially large. To do so, we fix the the set of sites where the particles eventually fall asleep, which reduces the problem to a simpler model with density one. Taking advantage of the Abelian property of the model, we show that the stabilization time stochastically dominates the escape time of a onedimensional random walk with a negative drift. We then check that this slow phase for the finite volume dynamics implies the existence of an active phase on the infinite lattice.
 Publication:

arXiv eprints
 Pub Date:
 March 2022
 DOI:
 10.48550/arXiv.2203.02476
 arXiv:
 arXiv:2203.02476
 Bibcode:
 2022arXiv220302476F
 Keywords:

 Mathematics  Probability;
 60K35 (Primary);
 82B26 (Secondary)
 EPrint:
 27 pages, new version with minor corrections