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).
Highlights
Let us consider a random process on a tree which is parametrized with one parameter p. We say that this process undergoes a phase transition if there exists a critical parameter pc such that the behavior of the random process is significantly different for p < pc and for p > pc
Bernoulli percolation and biased random walks share the same critical parameter which is equal to the branching number of the tree
In [8], it was proved that the critical parameter for the once-reinforced random walk on trees is equal to the branching-ruin number of the tree (see (2.2))
Summary
Let us consider a random process on a tree which is parametrized with one parameter p. Cookie random walk, heavy tailed distribution, recurrence, transience, branching number, branching-ruin number. In [8], it was proved that the critical parameter for the once-reinforced random walk on trees is equal to the branching-ruin number of the tree (see (2.2)). Even though our results hold for any branching-ruin number, for the sake of the following explanations, let us temporarily assume that b > 1, so that simple random walk is transient on this tree (see Theorem 2, or [8]). We obtain a much finer description of the process and we can prove that this random walk undergoes a phase transition on trees with polynomial gowth, i.e. on trees T where the branching-ruin number brr(T ) is finite. We refer to Theorem 5 for the more general result on the biased case and Theorem 7 for the case where the number of cookies on each vertex is inhomogeneous over the tree
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