On the trace of branching random walks

TL;DR

This paper investigates the properties of the trace of branching random walks on Cayley graphs, revealing its transience, percolation characteristics, exponential growth, and strong recurrence, with broader implications for unimodular random graphs.

Contribution

It establishes new results on the trace's transience, percolation threshold, volume growth, and recurrence, extending analysis to unimodular random graphs.

Findings

Trace is almost surely transient for simple random walk

Trace has critical percolation probability less than one

Trace exhibits exponential volume growth

Abstract

We study branching random walks on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for the simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any (non-trivial) branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All results are stated in the more general context of branching random walks on unimodular random graphs.