{
  "abstract": "We determine the monochromatic three-arm exponent for critical planar percolation through the exact spectral threshold of an explicit positive integral operator family with elementary kernels, obtaining $ξ_3=0.76928375\\cdots$. Unlike the monochromatic two-arm (backbone) event, the three-arm event can be obstructed by two distinct white clusters, so a Markovian exploration must retain the past connections. We introduce a two-point boundary observable that records these connections. Using the coupling between conformal loop ensemble (CLE) and Liouville quantum gravity (LQG), this observable satisfies a renewal equation in terms of quantum boundary lengths, and yields the explicit operator whose spectral information determines $ξ_3$. Our approach also gives an explicit spectral characterization of monochromatic $n$-arm exponents for all $n\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05101",
  "author_total": 1,
  "authors": [
    {
      "name": "Gefei Cai"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f951b5ma4rbr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05101",
  "fulltext_url": null,
  "id": "f951b5ma4rbr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05101",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f951b5ma4rbr9.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.05101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05101"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Monochromatic arm exponents for critical planar percolation",
  "title_text": "Monochromatic arm exponents for critical planar percolation",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/f951b5ma4rbr9",
  "venue": "arXiv",
  "versions": []
}