Non-intersection of transient branching random walks

TL;DR

This paper proves a conjecture that the traces of independent transient branching random walks on nonamenable Cayley graphs are almost surely disjoint, introducing the concept of local unimodularity with potential for broader applications.

Contribution

It confirms the conjecture that independent transient branching random walks on nonamenable groups do not intersect infinitely often, and introduces the new concept of local unimodularity.

Findings

Independent branching random walks do not intersect infinitely often.

The trace of the walk is almost surely tree-like with multiple ends.

Introduction of the concept of local unimodularity.

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 M\"uller (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 tree-like in the sense that it is infinitely-ended 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.