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

Andrea Collevecchio, Cong Bang Huynh, Daniel Kious

Research output: Contribution to journalArticleResearchpeer-review


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 languageEnglish
Article number121
Number of pages29
JournalElectronic Journal of Probability
Publication statusPublished - 1 Jan 2019


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

Cite this