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The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees
Author(s) -
Collevecchio Andrea,
Kious Daniel,
Sidoravicius Vladas
Publication year - 2020
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21860
Subject(s) - mathematics , random walk , branching (polymer chemistry) , combinatorics , loop erased random walk , tree (set theory) , heterogeneous random walk in one dimension , phase transition , discrete mathematics , class (philosophy) , branching random walk , statistical physics , statistics , computer science , materials science , physics , composite material , quantum mechanics , artificial intelligence
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. © 2019 Wiley Periodicals, Inc.