### Abstract

Original language | English |
---|---|

Pages (from-to) | 210-236 |

Number of pages | 27 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 73 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2020 |

### Cite this

*Communications on Pure and Applied Mathematics*,

*73*(1), 210-236. https://doi.org/10.1002/cpa.21860

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*Communications on Pure and Applied Mathematics*, vol. 73, no. 1, pp. 210-236. https://doi.org/10.1002/cpa.21860

**The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees.** / Collevecchio, Andrea; Kious, Daniel; Sidoravicius, Vladas.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - The Branching‐Ruin Number and the Critical Parameter of Once‐Reinforced Random Walk on Trees

AU - Collevecchio, Andrea

AU - Kious, Daniel

AU - Sidoravicius, Vladas

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85075090724&partnerID=8YFLogxK

U2 - 10.1002/cpa.21860

DO - 10.1002/cpa.21860

M3 - Article

VL - 73

SP - 210

EP - 236

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 1

ER -