### 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 |
---|---|

Article number | 121 |

Number of pages | 29 |

Journal | Electronic Journal of Probability |

Volume | 24 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

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

### Cite this

*Electronic Journal of Probability*,

*24*, [121]. https://doi.org/10.1214/19-EJP383

}

*Electronic Journal of Probability*, vol. 24, 121. https://doi.org/10.1214/19-EJP383

**The branching-ruin number as critical parameter of random processes on trees.** / Collevecchio, Andrea; Huynh, Cong Bang ; Kious, Daniel.

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

TY - JOUR

T1 - The branching-ruin number as critical parameter of random processes on trees

AU - Collevecchio, Andrea

AU - Huynh, Cong Bang

AU - Kious, Daniel

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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).

AB - 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).

KW - Branching number

KW - Branching-ruin number

KW - Cookie random walk

KW - Heavy tailed distribution

KW - Phase transition

KW - Random conductance model

UR - http://www.scopus.com/inward/record.url?scp=85075612476&partnerID=8YFLogxK

U2 - 10.1214/19-EJP383

DO - 10.1214/19-EJP383

M3 - Article

AN - SCOPUS:85075612476

VL - 24

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

M1 - 121

ER -