The branching-ruin number and the critical parameter of once-reinforced random walk on trees

Andrea Collevecchio, Daniel Kious, Vladas Sidoravicius

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Abstract

The motivation for this paper is the study of the phase transition for recurrence/ transience of a class of self‐interacting random walks on trees, which includes the once‐reinforced random walk. For this purpose, we define a quantity, which we call the branching‐ruin number of a tree, which provides (in the spirit of Furstenberg [11] and Lyons [13]) a natural way to measure trees with polynomial growth. We prove that the branching‐ruin number of a tree is equal to the critical parameter for the recurrence/transience of the once‐reinforced random walk. We define a sharp and effective (i.e., computable) criterion characterizing the recurrence/transience of a larger class of self‐interacting walks on trees, providing the complete picture for their phase transition.
Original languageEnglish
Pages (from-to)210-236
Number of pages27
JournalCommunications on Pure and Applied Mathematics
Volume73
Issue number1
Early online date2 Aug 2019
DOIs
Publication statusPublished - 12 Nov 2019

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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