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  and Lyons ) 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.
|Number of pages||27|
|Journal||Communications on Pure and Applied Mathematics|
|Early online date||2 Aug 2019|
|Publication status||Published - 12 Nov 2019|
ASJC Scopus subject areas
- Applied Mathematics
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