2020
DOI: 10.48550/arxiv.2009.09491
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Active Phase for Activated Random Walk on Z

Abstract: We consider the Activated Random Walk model on Z. In this model, each particle performs a continuoustime simple symmetric random walk, and falls asleep at rate λ. A sleeping particle does not move but it is reactivated in the presence of another particle. We show that for any sleep rate λ < ∞ if the density ζ is close enough to 1 then the system stays active.

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Cited by 3 publications
(6 citation statements)
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“…We conclude by stating two conjectures. We remark that the inequality ζ c < 1 has been proved in dimensions d ≥ 3 by Taggi [29] and in dimension 1 by Hoffman, Richey, and Rolla [10]. It remains open in dimension 2.…”
Section: Conjecturesmentioning
confidence: 70%
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“…We conclude by stating two conjectures. We remark that the inequality ζ c < 1 has been proved in dimensions d ≥ 3 by Taggi [29] and in dimension 1 by Hoffman, Richey, and Rolla [10]. It remains open in dimension 2.…”
Section: Conjecturesmentioning
confidence: 70%
“…The c 1 and c 2 above are absolute constants; the proof will show that c 1 = 1 41 and c 2 = 1 5 suffice. Combining Theorem 4 with the bound (10), we obtain an upper bound on the mixing time of the ARW process.…”
Section: Theorem 4 (Upper Bound For the Fill Time Of Idla)mentioning
confidence: 87%
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“…The approach from [2] was later reformulated in [6] as based on two major ingredients: a mass balance equation between blocks and a single-block estimate. More recently, this reformulation has been adopted successfully in [1,5] to find the right order of asymptotics in Corollary 1.3 and extend Corollary 1.2 for all λ > 0 respectively. All these works rely crucially on the topological convenience of Z in order to maintain independence structures between blocks.…”
Section: Introductionmentioning
confidence: 99%