{
  "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 $\\mathbb Z^d$, and we estimate the $α$-stable branching capacity of balls.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30668",
  "author_total": 1,
  "authors": [
    {
      "name": "Antoine Aurillard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dgaqssjxfd1gz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30668",
  "fulltext_url": null,
  "id": "dgaqssjxfd1gz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30668",
  "language": "en",
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7204877011",
  "pdf_url": "https://arxiv.org/pdf/2608.30668",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dgaqssjxfd1gz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30668",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30668"
    },
    {
      "first_seen": "2026-10-30T12:17:53Z",
      "native_id": "hal-05789929",
      "source": "hal",
      "url": "https://hal.science/hal-05789929v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Recurrence and capacity of stable branching random walks",
  "title_text": "Recurrence and capacity of stable branching random walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:53Z",
  "url": "https://research.draconitic.com/paper/dgaqssjxfd1gz",
  "venue": "arXiv",
  "versions": []
}