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PreprintCC BY-SA 4.0arXiv

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 μ is critical and attracted to an α-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 α-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 ℤd, and we estimate the α-stable branching capacity of balls.

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Cite

BibTeXRIS
BibTeX
@misc{aurillard2026recurrence,
  title = {{Recurrence and capacity of stable branching random walks}},
  author = {Antoine Aurillard},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30668},
  eprint = {2608.30668},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30668},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Recurrence and capacity of stable branching random walks
AU  - Antoine Aurillard
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30668
UR  - https://arxiv.org/abs/2608.30668
AB  - We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution μ is critical and attracted to an α-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 α-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 ℤ^d, and we estimate the α-stable branching capacity of balls.
ER  -