Separation cutoff for activated random walks
Bristiel, Alexandre; Salez, Justin (2022), Separation cutoff for activated random walks. https://basepub.dauphine.psl.eu/handle/123456789/23735
TypeDocument de travail / Working paper
Series titleCahier de recherche CEREMADE, Université Paris Dauphine-PSL
MetadataShow full item record
École normale supérieure de Lyon [ENS de Lyon]
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We consider Activated Random Walks on arbitrary finite networks, with particles being inserted at random and absorbed at the boundary. Despite the non-reversibility of the dynamics and the lack of knowledge on the stationary distribution, we explicitly determine the relaxation time of the process, and prove that separation cutoff is equivalent to the product condition. We also provide sharp estimates on the center and width of the cutoff window. Finally, we illustrate those results by establishing explicit separation cutoffs on various networks, including: (i) large finite subgraphs of any fixed infinite non-amenable graph, with absorption at the boundary and (ii) large finite vertex-transitive graphs with absorption at a single vertex. The latter result settles a conjecture of Levine and Liang. Our proofs rely on the refined analysis of a strong stationary time recently discovered by Levine and Liang and involving the IDLA process.
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