{
  "abstract": "We consider a branching random walk on $\\mathbb{Z}^2$ with critical offspring of mean $1$ and spatial motion governed by the jumps of a lazy simple random walk. For every site $x\\in\\mathbb{Z}^2$, we obtain a uniform asymptotical estimate for the local survival probability, i.e., the probability that there are particles at $x$ at large time $n$. Using Stein's method, we establish a Yaglom-type theorem for the number of particles at $x$ at time $n$ when $x$ is at distance of order $\\sqrt{n}$ from the origin. Moreover, at the position occupied by a typical particle at time $n$, the number of particles at that site, normalized by $\\log n$, converges in law to a Gamma distribution, thereby confirming a conjecture of Lalley and Zheng [Ann. Probab. 39 (2010), 327-368]. Finally, we prove that, conditional on local survival, the total number of particles at time $n$, divided by $n$, also converges weakly to a Gamma distribution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15538",
  "author_total": 3,
  "authors": [
    {
      "name": "Tianyi Bai"
    },
    {
      "name": "Xinxin Chen"
    },
    {
      "name": "Shen Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/918b1a2zzdgj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15538",
  "fulltext_url": null,
  "id": "918b1a2zzdgj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15538",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213313565",
  "pdf_url": "https://arxiv.org/pdf/2609.15538",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/918b1a2zzdgj1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.15538",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15538"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Critical branching random walks on $\\mathbb{Z}^2$: local survival probabilities and Yaglom limit theorems",
  "title_text": "Critical branching random walks on ℤ^2: local survival probabilities and Yaglom limit theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/918b1a2zzdgj1",
  "venue": "arXiv",
  "versions": []
}