## Abstract

The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees. In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees. First, we study random walks on random conductances on trees, when the conductances have an heavy tail at 0, parametrized by some p > 1, where 1/p is the exponent of the tail. We prove a phase transition recurrence/transience with respect to p and identify the critical parameter to be equal to the branching-ruin number of the tree. Second, we study a multi-excited random walk on trees where each vertex has M cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).

Original language | English |
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Article number | 121 |

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Electronic Journal of Probability |

Volume | 24 |

Early online date | 6 Nov 2019 |

DOIs | |

Publication status | Published - 6 Nov 2019 |

### Bibliographical note

Funding Information:DK is grateful to NYU Shanghai, where he was affiliated to at the time this work was initiated.

Publisher Copyright:

© 2019, Institute of Mathematical Statistics. All rights reserved.

## Keywords

- Branching number
- Branching-ruin number
- Cookie random walk
- Heavy tailed distribution
- Phase transition
- Random conductance model

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty