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Non-intersection of transient branching random walks

Hutchcroft, Tom (2019) Non-intersection of transient branching random walks. . (Unpublished)

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Let G be a Cayley graph of a nonamenable group with spectral radius ρ<1. It is known that branching random walk on G with offspring distribution μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring μ¯ satisfies μ¯≤ρ−1. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase 1<μ¯≤ρ−1, and in particular at the recurrence threshold μ¯=ρ−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.

Item Type:Report or Paper (Discussion Paper)
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Hutchcroft, Tom0000-0003-0061-593X
Additional Information:We thank Itai Benjamini, Jonathan Hermon, Asaf Nachmias, and Elisabetta Candellero for useful discussions. In particular, we thank Asaf for discussions that led to a substantially simpler proof of Theorem 3.3. We also thank the anonymous referee for their careful reading and helpful suggestions.
Record Number:CaltechAUTHORS:20210924-202112742
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111030
Deposited By: George Porter
Deposited On:27 Sep 2021 16:51
Last Modified:27 Sep 2021 16:51

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