English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

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 smallenough, 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 sleeprate, the stabilization time of the model on the $d$-dimensional torus isexponentially large. To do so, we fix the the set of sites where the particleseventually fall asleep, which reduces the problem to a simpler model withdensity one. Taking advantage of the Abelian property of the model, we showthat the stabilization time stochastically dominates the escape time of aone-dimensional random walk with a negative drift. We then check that this slowphase for the finite volume dynamics implies the existence of an active phaseon the infinite lattice.

Active Phase for Activated Random Walks on the Lattice in all Dimensions