Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity

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 (Invent. Math. 188 (2012) 127–150). 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.