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Published October 2020 | Published + Accepted Version
Journal Article Open

Non-intersection of transient branching random walks


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[bar] satisfies μ[bar] ≤ ρ − 1. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase 1< μ[bar] ≤ ρ − 1, and in particular at the recurrence threshold μ[bar] = 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.

Additional Information

© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Received: 7 October 2019 / Revised: 6 February 2020 / Published online: 25 February 2020. 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.

Attached Files

Published - Hutchcroft2020_Article_Non-intersectionOfTransientBra.pdf

Accepted Version - 1910.01018.pdf


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