Let G be a Cayley graph of a nonamenable group with spectral radius $$\rho < 1$$ ρ < 1 . It is known that branching random walk on G with offspring distribution $$\mu $$ μ is transient , i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $${\overline{\mu }}$$ μ ¯ satisfies $$\overline{\mu }\le \rho ^{-1}$$ μ ¯ ≤ ρ - 1 . Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase $$1<\overline{\mu } \le \rho ^{-1}$$ 1 < μ ¯ ≤ ρ - 1 , and in particular at the recurrence threshold $${\overline{\mu }} = \rho ^{-1}$$ 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.

中文翻译：

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瞬态分支随机游走的非交集

设 G 是谱半径 $$\rho < 1$$ ρ < 1 的不可命名群的凯莱图。已知具有后代分布 $$\mu $$ μ 的 G 上的分支随机游走是瞬态的，即，当且仅当后代的预期数量 $${\overline{\ mu }}$$ μ¯ 满足 $$\overline{\mu }\le \rho ^{-1}$$ μ¯ ≤ ρ - 1 。Benjamini 和 Müller (Groups Geom Dyn, 6:231–247, 2012) 推测在整个瞬态超临界相 $$1<\overline{\mu} \le \rho ^{-1}$$ 1 < μ¯ ≤ ρ - 1 ，特别是在递归阈值 $${\overline{\mu }} = \rho ^{-1}$$ μ¯ = ρ - 1 ，分支随机游走的轨迹在某种意义上是树状的如果步行永远存在，它几乎肯定会无限结束。这本质上等同于分支随机游走的两个独立副本最多有限地经常几乎肯定地相交的断言。我们证明了这个猜想，以及由同一作者做出的其他几个相关猜想。这项工作的一个核心贡献是引入了局部单模性的概念，我们希望在未来有几个进一步的应用。