# Groups, Geometry, and Dynamics

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**Volume 6, Issue 2, 2012, pp. 231–247**

**DOI: 10.4171/GGD/156**

Published online: 2012-04-16

On the trace of branching random walks

Itai Benjamini^{[1]}and Sebastian Müller

^{[2]}(1) Weizmann Institute of Science, Rehovot, Israel

(2) Université de Provence, Marseille, France

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 almost surely (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.

*Keywords: *Branching random walk, trace, unimodular random network, recurrence, invariant percolation

Benjamini Itai, Müller Sebastian: On the trace of branching random walks. *Groups Geom. Dyn.* 6 (2012), 231-247. doi: 10.4171/GGD/156