Recurrence and capacity of stable branching random walks

Antoine Aurillard

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arXiv · 2608.30668

Recurrence and capacity of stable branching random walks

Antoine Aurillard

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Abstract

We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $\mu$ is critical and attracted to an $\alpha$-stable law. This is done via the infinite invariant BRW introduced by Le Gall & Lin and a new criterion which relates this growth rate of the range to a notion of dimension of the underlying tree in a general way. Then, in the transient case (that is, when the range does grow linearly), we extend the notion of branching capacity to this $\alpha$-stable case. We show that it is still related to the asymptotic probability that a BRW (or its infinite version) reaches a distant set in $\mathbb Z^d$, and we estimate the $\alpha$-stable branching capacity of balls.

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BibTeX RIS

Antoine Aurillard. 2026-08-31. Recurrence and capacity of stable branching random walks. https://arxiv.org/abs/2608.30668

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