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 (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.
Acknowledgments
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.
Citation
Lorenzo Taggi. "Essential enhancements in Abelian networks: Continuity and uniform strict monotonicity." Ann. Probab. 51 (6) 2243 - 2264, November 2023. https://doi.org/10.1214/23-AOP1647
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