WIAS Preprint No. 2722, (2020)

# Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity

Authors

• Taggi, Lorenzo

2010 Mathematics Subject Classification

• 82C22 60K35 82C26

Keywords

• Essential enhancements, activated random walks, Abelian networks, self-organised criticality, absorbing-state phase transition

DOI

10.20347/WIAS.PREPRINT.2722

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.