{
  "abstract": "We study clusters of the Brownian loop soup on $\\mathbb{R}^3$ and of random walk and metric-graph loop soups on $\\mathbb{Z}^d$ and $\\widetilde{\\mathbb{Z}}^d$ within a unified framework. For the continuum model, when the one-arm exponent $ξ(α)$ is positive, we prove that almost surely every cluster has Hausdorff dimension $3-ξ(α)$. A key step is to show that, conditioned on the one-arm event, a crossing cluster has capacity comparable to the crossing scale with uniformly positive probability. This gives one-arm estimates of order $αr^{-ξ(α)}$, uniformly down to zero intensity, and shows that $ξ(α)$ is strictly decreasing and left-continuous. Moreover, $1-ξ(α)\\asympα$ as $α\\downarrow0$. As another application of the one-arm estimates, we prove that several critical values coincide, resolving a conjecture of Jego and Lupu in arXiv:2601.04840. The proof combines a decoupling technique with a multiscale recursive argument to control the combined effects of large and small loops. These methods also extend to loop soups on $\\mathbb{Z}^d$ and $\\widetilde{\\mathbb{Z}}^d$ for $d=3,4,5$, yielding capacity lower bounds of order $r$, $r^2/\\log r$, and $r^2$ for the cluster of the origin, conditioned on the one-arm event, throughout the small-loop decay regime. In particular, these bounds imply strict polynomial comparisons between one-arm probabilities at different intensities in dimension three. We also show that the small-loop decay regime on $\\widetilde{\\mathbb{Z}}^d$ has upper endpoint $\\frac{1}{2}$ by adapting the OSSS sharpness framework to the truncated loop soup.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38510",
  "author_total": 1,
  "authors": [
    {
      "name": "Yifan Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bjtq8z5g6k5az.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38510",
  "fulltext_url": null,
  "id": "bjtq8z5g6k5az",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38510",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38510",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bjtq8z5g6k5az.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38510",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38510"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "On loop-soup clusters in three dimensions and higher",
  "title_text": "On loop-soup clusters in three dimensions and higher",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/bjtq8z5g6k5az",
  "venue": "arXiv",
  "versions": []
}