On the slow phase for fixed-energy Activated Random Walks

TL;DR

This paper proves that the Activated Random Walk model on a 1D ring exhibits a slow phase at high density, with activity persisting for a long time regardless of sleep rates.

Contribution

It introduces a new toppling procedure to demonstrate the existence of a slow phase in the model at high density, regardless of sleep rates.

Findings

Activity persists for a long time in the high-density regime.

The slow phase exists for arbitrarily large sleep rates.

A new proof technique is developed for this phenomenon.

Abstract

We study the Activated Random Walk model on the one-dimensional ring, in the high density regime. We develop a toppling procedure that gradually builds an environment that can be used to show that activity will be sustained for a long time. This yields a self-contained and relatively short proof of existence of a slow phase for arbitrarily large sleep rates.