{
  "abstract": "Let $K_1=B[0,1]$ be the trace of a planar Brownian motion, and let $N_{\\mathrm{in}}(η)$ count the bounded components of $\\mathbb{R}^2\\setminus K_1$ with inradius greater than $η$. As $η\\downarrow0$, we prove that $η^2(\\log(1/η))^2N_{\\mathrm{in}}(η)\\to c_{\\mathrm{in}}=(π/2)\\int G^2d\\mathcal{L}\\in(0,1/2]$ almost surely and in $L^p$ for every $p\\in[1,2]$, where $G$ is the inradius divided by the square root of the area and $\\mathcal{L}$ is Werner's limiting shape law. Inradius determines survival under thickening: a cavity contains a hole of the Wiener sausage of radius $r$ if and only if its inradius exceeds $r$. Brownian scaling shows that, for fixed $r>0$ as $T\\to\\infty$, the number of cavities of $B[0,T]$ with inradius greater than $r$ is asymptotic to $4c_{\\mathrm{in}}T/(r^2(\\log T)^2)$ in probability and in $L^p$, $p\\in[1,2]$. It also gives lower bounds for the sausage's hole count, the Betti curve of its degree-one persistent homology, and its radius integrals, with constant $4c_{\\mathrm{in}}$ in expectation and any smaller constant with probability tending to one. A matching upper bound cannot follow from general bounds in terms of perimeter, area, diameter or length. We prove that it is equivalent to a first-moment estimate for near-critical pockets: holes at level $η$ on which the distance to the trace is at most $2η$. Equivalently, it suffices to control residual pockets beyond one per cavity with inradius in $(η,2η]$. This estimate is proved in a companion paper, yielding the exact order $T/(r^2(\\log T)^2)$ for the expected number of holes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.11248",
  "author_total": 1,
  "authors": [
    {
      "name": "Tristan Guillaume"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35zwzsej26210.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.11248",
  "fulltext_url": null,
  "id": "35zwzsej26210",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.11248",
  "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",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7164369388",
  "pdf_url": "https://arxiv.org/pdf/2606.11248",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35zwzsej26210.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2606.11248",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.11248"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage",
  "title_text": "The inradius law for the cavities of planar Brownian motion, with applications to the Wiener sausage",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:42Z",
  "url": "https://research.draconitic.com/paper/35zwzsej26210",
  "venue": "arXiv",
  "versions": []
}