Essential enhancements in Abelian networks: continuity and uniform strict monotonicity

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.

Exact sampling and fast mixing of Activated Random Walk

“…Recent work has succeeded in showing that ζ c is strictly between 0 and 1 on many transitive graphs [28], including Z d for d ≥ 3 [29] and d = 1 [10]; that ζ c is continuous and strictly increasing in the sleep rate [30], and tends to zero as the square root of the sleep rate [3]; and that ARW at sufficiently high density takes exponential time to stabilize on a cycle [4]. Despite all this progress, very little is known about the behavior of Activated Random Walk at criticality.…”