{
  "abstract": "In the random Lorentz gas, a point tracer moves through spherical obstacles of (reduced) density $Φ$ centered on a Poisson point process in $d$-dimensional space, $\\mathbb{R}^d$, and becomes localized once the vacant space stops percolating, at $Φ_{\\rm p}(d)$. The dimensional scaling of this void percolation threshold has remained elusive. Prior bounds, conjectural ones included, ranged from $Ω(1)$ to $O(d\\ln d)$, leaving even the scaling undetermined. We here prove that $Φ_{\\rm p}(d)=Θ(d)$, and in particular that $\\tfrac12\\,d \\le Φ_{\\rm p}(d) \\le 5\\,d$ for all sufficiently large $d$. The upper bound counts hierarchical families of escape waypoints called tree embeddings; the lower bound is via a weighted second-moment estimate on an explicit ensemble of ballistic escape routes. A linear scaling of $Φ_{\\rm p}(d)$ excludes the $d\\ln d$ growth suggested by analogy with other systems in the random first-order transition universality class and is consistent with the numerical estimate $Φ_{\\rm p}/d \\to 3.40(5)$. Because the mean-field dynamical transition is likewise linear, $Φ_{\\rm d}\\simeq 2.40\\,d$, our result fixes the activated window $[Φ_{\\rm d},Φ_{\\rm p}]$ as a bounded---rather than diverging---correction to mean-field dynamical arrest in sufficiently high $d$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07196",
  "author_total": 2,
  "authors": [
    {
      "name": "Patrick Charbonneau"
    },
    {
      "name": "Marcus Michelen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8dk34bjqh847d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07196",
  "fulltext_url": null,
  "id": "8dk34bjqh847d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07196",
  "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",
      "cond-mat.dis-nn"
    ],
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07196",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8dk34bjqh847d.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:30Z",
      "native_id": "2611.07196",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07196"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.dis-nn",
      "subfield": "condensed-matter"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "On the critical value of void percolation in high dimensions",
  "title_text": "On the critical value of void percolation in high dimensions",
  "type": "preprint",
  "updated_at": "2026-11-08T21:20:42Z",
  "url": "https://research.draconitic.com/paper/8dk34bjqh847d",
  "venue": "arXiv",
  "versions": []
}