Non-intersection of transient branching random walks
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Non-intersection of transient branching random walks Tom Hutchcroft1 Received: 7 October 2019 / Revised: 6 February 2020 © The Author(s) 2020
Abstract 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 (Groups Geom Dyn, 6:231–247, 2012) 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. Mathematics Subject Classification 60D04 · 60K35
1 Introduction Let G = (V , E) be a connected, locally finite graph. Branching random walk on G is a Markov process taking values in the space of finitely-supported functions V → {0, 1, 2, . . .}, which we think of as encoding the number of particles occupying each vertex of G. We begin with a single particle, which occupies some vertex v. At every time step, each particle splits into a random number of new particles according to a fixed offspring distribution μ, each of which immediately performs a simple random walk step on G. Equivalently, branching random walk can be described as a random walk on G indexed by a Galton-Watson tree [10,11]. We say that the offspring distribution μ is non-trivial if μ(1) < 1. It follows from the classical theory of
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Tom Hutchcroft [email protected] Statslab, DPMMS, University of Cambridge, Cambridge, UK
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T. Hutchcroft
branching processes (see e.g. [42, Chapter 5]) that branching random walk exhibits a phase transition: If the mean offspring μ satisfies μ > 1 then the process survives forever with positive probability, while if μ is non-trivial and μ ≤ 1 then the process survives for only finitely many time steps almost surely. Beyond its intrinsic appeal and its function as a model for many processes appearing in the natural sciences, branching random walk also attracts attention as a toy model that lends insight into more complex processes. Indeed, many models of statistical mechanics are expected to have mean-field behaviour in high dimensions, which roughly means that their behaviour at criticality is similar to that of a critical branching random walk. Mean-field behaviour has now been proven to hold in high dimensions for percolation [32,33], the Ising model [1], the contact process [46], uniform spanning trees [35,45], and t
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