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The branching-ruin number of a tree, which describes its asymptotic growthand geometry, can be seen as a polynomial version of the branching number. Thisquantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order tounderstand the phase transitions of the once-reinforced random walk (ORRW) ontrees. Strikingly, this number was proved to be equal to the critical parameterof ORRW on trees. In this paper, we continue the investigation of the linkbetween the branching-ruin number and the criticality of random processes ontrees. First, we study random walks on random conductances on trees, when theconductances 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/transiencewith respect to $p$ and identify the critical parameter to be equal to thebranching-ruin number of the tree. Second, we study a multi-excited random walkon trees where each vertex has $M$ cookies and each cookie has an infinitestrength towards the root. Here again, we prove a phase transitionrecurrence/transience and identify the critical number of cookies to be equalto the branching-ruin number of the tree, minus 1. This result extends aconjecture of Volkov (2003). Besides, we study a generalized version of thisprocess and generalize results of Basdevant and Singh (2009).

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