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

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 branchingruin 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).

The branching number of intermediate growth trees

“…Random walks with random conductance. In [CHK19] the authors study a random walk in heavy-tailed random conductances on polynomial-growth trees. They prove a phase transition between recurrence and transience depending on the tail of the law of the conductances, and show that the branching-ruin number is the critical threshold for this transition.…”

“…In this section, we follow the blueprint in [CHK19] to prove Theorem 2.5. We define ψ RC : Ψ(e) γ > 0 , (7.3)…”

“…One limitation of the branching number is that it is always 1 for subexponential growth trees, thus it does not convey any information on them. A step in generalizing the branching number to smaller trees was done in [CHK18,CHK19] where an analogous branchingruin number on polynomial growth trees was introduced and studied. While it managed to capture threshold properties for several random processes on polynomial growth trees (e.g.…”