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2019 The branching-ruin number as critical parameter of random processes on trees
Andrea Collevecchio, Cong Bang Huynh, Daniel Kious
Electron. J. Probab. 24(none): 1-29 (2019). DOI: 10.1214/19-EJP383

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).

Citation

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Andrea Collevecchio. Cong Bang Huynh. Daniel Kious. "The branching-ruin number as critical parameter of random processes on trees." Electron. J. Probab. 24 1 - 29, 2019. https://doi.org/10.1214/19-EJP383

Information

Received: 30 September 2019; Accepted: 27 October 2019; Published: 2019
First available in Project Euclid: 6 November 2019

zbMATH: 07142915
MathSciNet: MR4029424
Digital Object Identifier: 10.1214/19-EJP383

Subjects:
Primary: 60K35, 60K37, 82D30

JOURNAL ARTICLE
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