Nonintersection of transient branching random walks
Abstract
Let $G$ be a Cayley graph of a nonamenable group with spectral radius $\rho < 1$. It is known that branching random walk on $G$ with offspring distribution $\mu$ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $\bar \mu$ satisfies $\bar \mu \leq \rho^{1}$. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase $1<\bar{\mu} \leq \rho^{1}$, and in particular at the recurrence threshold $\bar \mu = \rho^{1}$, the trace of the branching random walk is treelike in the sense that it is infinitelyended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 DOI:
 10.48550/arXiv.1910.01018
 arXiv:
 arXiv:1910.01018
 Bibcode:
 2019arXiv191001018H
 Keywords:

 Mathematics  Probability;
 Mathematics  Group Theory
 EPrint:
 22 pages V2: Several minor corrections and improvements. Accepted version, to appear in PTRF