Once reinforced random walk on Z×Gamma

Daniel Kious, Bruno Schapira, Arvind Singh

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Abstract

We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form Z×Γ, with Γ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.
Original languageEnglish
Pages (from-to)2219-2242
Number of pages23
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume57
Issue number4
Early online date20 Oct 2021
DOIs
Publication statusPublished - 30 Nov 2021

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