November 2023 Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity
Lorenzo Taggi
Author Affiliations +
Ann. Probab. 51(6): 2243-2264 (November 2023). DOI: 10.1214/23-AOP1647


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.

Funding Statement

The author acknowledges support from DFG German Research Foundation BE/5267/1 and from EPSRC Early Career Fellowship EP/N004566/1.


This work started as the author was affiliated to Technische Universität Darmstadt, it has been carried on while the author was affiliated to the University of Bath, it was concluded as the author was affiliated to the Weierstrass Institute, Berlin and it was revised as the author was affiliated to Sapienza Università di Roma. The author thanks the two anonymous referees for carefully reviewing the paper and their important and useful comments.


Download Citation

Lorenzo Taggi. "Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity." Ann. Probab. 51 (6) 2243 - 2264, November 2023.


Received: 1 September 2022; Revised: 1 July 2023; Published: November 2023
First available in Project Euclid: 12 November 2023

Digital Object Identifier: 10.1214/23-AOP1647

Primary: 60K35 , 82C22
Secondary: 82C26

Keywords: Abelian networks , Absorbing-state phase transition , Activated Random Walks , essential enhancements , self-organised criticality

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.51 • No. 6 • November 2023
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