Electronic Journal of Probability

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

Andrea Collevecchio, Cong Bang Huynh, and Daniel Kious

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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).

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 121, 29 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1573030842

Digital Object Identifier
doi:10.1214/19-EJP383

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
random conductance model cookie random walk heavy tailed distribution phase transition branching number branching-ruin number

Rights
Creative Commons Attribution 4.0 International License.

Citation

Collevecchio, Andrea; Huynh, Cong Bang; Kious, Daniel. The branching-ruin number as critical parameter of random processes on trees. Electron. J. Probab. 24 (2019), paper no. 121, 29 pp. doi:10.1214/19-EJP383. https://projecteuclid.org/euclid.ejp/1573030842


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