Once reinforced random walk on Z × Γ

Daniel Kious, Bruno Schapira, Arvind Singh

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We revisit Vervoort’s unpublished paper (Vervoort (2002)) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form Z × Γ, with Γ a finite graph, for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 pages24
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number4
Early online date20 Oct 2021
Publication statusPublished - 30 Nov 2021


  • Recurrence
  • Reinforced random walk
  • Self-interacting random walk
  • Shape theorem

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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