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 {overline{mu }} satisfies overline{mu }le rho ^{-1}. Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase 1<overline{mu } le rho ^{-1}, and in particular at the recurrence threshold {overline{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.
Highlights
Let G = (V, E) be a connected, locally finite graph
We briefly recall the definition of unimodular random rooted graphs and some basic facts about them
Together with a distinguished vertex u, the root. (We will often use the convention of using lower case letters for deterministic rooted graphs and upper case letters for random rooted graphs.) An isomorphism of graphs is an isomorphism of rooted graphs if it preserves the root
Summary
When G is a transitive nonamenable graph, such as a Cayley graph of a nonamenable group, rather more is known: For μ > P −1 the branching random walk visits every vertex infinitely often almost surely on the event that it survives forever [11, Lemma 5.1], while for μ ≤ P −1 the expected number of times the walk returns to the origin is finite [48, Theorem 7.8]. Using our formulation of the Magic Lemma together with the local unimodularity result Proposition 3.1, we are able to prove that the set I is either finite or accumulates to at most two ends of T almost surely The latter possibility is ruled out using the Markovian nature of branching random walk, completing the proof. The details of these generalizations are straightforward, and we restrict attention to the above case for clarity of exposition
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