{
  "abstract": "We study collision times of many independent simple random walks on $\\mathbb Z^2$ through the joint moment generating function of their pairwise collision local times. For a fixed number of walks, these collision times are known to be asymptotically independent after a suitable logarithmic normalisation. We investigate the extent to which this independence persists when the number of walks grows. For $N$ being the walk length, our results identify a threshold $\\asymp (\\log N)^{\\frac{1}{3}}$, on which the transition from independence to dependence happens. The proofs combine chaos expansion techniques and a correlation inequality, which is the result of a local limit theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21937",
  "author_total": 1,
  "authors": [
    {
      "name": "Ziyang Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77wpks5ty1zx0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21937",
  "fulltext_url": null,
  "id": "77wpks5ty1zx0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21937",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213933789",
  "pdf_url": "https://arxiv.org/pdf/2609.21937",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77wpks5ty1zx0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.21937",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21937"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Independence Threshold for Collision Times of Many Planar Random Walks",
  "title_text": "Independence Threshold for Collision Times of Many Planar Random Walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/77wpks5ty1zx0",
  "venue": "arXiv",
  "versions": []
}