Essential enhancements in Abelian networks: continuity and uniform strict monotonicity

TL;DR

This paper establishes the continuity and strict monotonicity of the critical curve in activated random walk models, extending previous results and introducing a novel proof method based on essential enhancements in Abelian networks.

Contribution

It introduces a new proof technique for Abelian networks that proves continuity and strict monotonicity of the critical curve, generalizing prior results.

Findings

Critical curve is continuous with respect to deactivation rate.

Strict monotonicity properties for increasing events are established.

A new proof method based on essential enhancements is developed.

Abstract

We prove that in wide generality the critical curve of the activated random walk model is a continuous function of the deactivation rate, and we provide a bound on its slope which is uniform with respect to the choice of the graph. Moreover, we derive strict monotonicity properties for the probability of a wide class of `increasing' events,extending previous results of Rolla and Sidoravicius (2012). Our proof method is of independent interest and can be viewed as a reformulation of the `essential enhancements' technique -- which was introduced for percolation -- in the framework of Abelian networks.