{
  "abstract": "We study the critical Brownian loop soup on the metric graph $\\widetilde{\\mathbb Z}^d$. Using a new unified approach, we determine the one-arm probability up to multiplicative constants in all dimensions $d\\ge3$. In particular, at the upper critical dimension $d_c=6$, we show that the one-arm probability is comparable to $r^{-2}(\\log r)^{\\frac12}$, and moreover, with high probability a box of radius $r$ contains on the order of $\\log r$ macroscopic clusters, each with volume of order $r^4(\\log r)^{-\\frac12}$. For $d\\ne6$, we recover the one-arm estimates of Cai and Ding, $r^{-(d-2)/2}$ for $d=3,4,5$ and $r^{-2}$ for $d>6$ (arXiv:2406.02397; Probab. Theory Relat. Fields, 2025), and the three-dimensional estimate of Drewitz, Prévost and Rodriguez (Probab. Theory Relat. Fields, 2026). Our proof adapts the entropy method of Dewan and Muirhead (Probab. Theory Relat. Fields, 2023) to the loop-soup setting. This yields a renormalization inequality for the one-arm probability, from which we derive the upper bounds. Independently, Rodriguez, Shi, and Zhang (arXiv:2611.05357) have recently used a closely related entropy approach to recover the one-arm estimates for $d>6$ and improve the upper bound for $d=6$. They also announce the same sharp one-arm estimate for $d=6$ in forthcoming work.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06298",
  "author_total": 1,
  "authors": [
    {
      "name": "Yifan Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fm3a50hvph4gt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06298",
  "fulltext_url": null,
  "id": "fm3a50hvph4gt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06298",
  "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/2611.06298",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fm3a50hvph4gt.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06298",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06298"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Critical one-arm probabilities for metric-graph loop soups in dimensions $d\\ge3$",
  "title_text": "Critical one-arm probabilities for metric-graph loop soups in dimensions d≥3",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/fm3a50hvph4gt",
  "venue": "arXiv",
  "versions": []
}