{
  "abstract": "We prove that the minimum Hausdorff dimension of a continuous curve joining $W_0$ to $W_1$ within the range $W([0,1])$ of a three-dimensional Brownian motion is almost surely a deterministic constant $γ_3\\in(1,5/3]$. Moreover, we construct continuum many simple curves with pairwise distinct images that attain this minimum. Each curve has box dimension $γ_3$ and admits a parametrization that is $β$-Hölder for every $0<β<1/γ_3$. To identify $γ_3$, we partition Brownian time into $n$ equal intervals and form a graph by joining pairs with intersecting ranges. We show that its endpoint graph distance and diameter have a common deterministic almost sure growth exponent $θ$ and that $γ_3=2θ$. We also establish $n^{pθ+o(1)}$ growth for their $p$-th moments, for every $p>0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32137",
  "author_total": 1,
  "authors": [
    {
      "name": "Daisuke Shiraishi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3krp3m60rvxbg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32137",
  "fulltext_url": null,
  "id": "3krp3m60rvxbg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32137",
  "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": "W7223702147",
  "pdf_url": "https://arxiv.org/pdf/2610.32137",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3krp3m60rvxbg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32137",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32137"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Graph distance and percolation dimension of three-dimensional Brownian motion",
  "title_text": "Graph distance and percolation dimension of three-dimensional Brownian motion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/3krp3m60rvxbg",
  "venue": "arXiv",
  "versions": []
}