{
  "abstract": "Fix $0<κ<8$ and put $d=1+κ/8$. Recent work by OpenAI identifies an explicit exact Hausdorff gauge \\( h_κ(r)=r^d\\bigl(\\log\\log(1/r)\\bigr)^{(2-d)/2} \\) for sufficiently small $r$ for chordal $\\mathrm{SLE}_κ$: the corresponding Hausdorff measure is almost surely positive and finite on every segment $γ([s,t])$ with $0<s<t<\\infty$, and has finite expectation in bounded subsets of the plane. We show that this Hausdorff measure is a deterministic positive constant multiple of the occupation measure (natural length) constructed by Lawler and Rezaei.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15066",
  "author_total": 1,
  "authors": [
    {
      "name": "Dapeng Zhan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/68p5775c3qh2s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15066",
  "fulltext_url": null,
  "id": "68p5775c3qh2s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15066",
  "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": "W7221801326",
  "pdf_url": "https://arxiv.org/pdf/2610.15066",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/68p5775c3qh2s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15066",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15066"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An Exact Hausdorff Measure as Natural Length for SLE",
  "title_text": "An Exact Hausdorff Measure as Natural Length for SLE",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/68p5775c3qh2s",
  "venue": "arXiv",
  "versions": []
}