{
  "abstract": "We study the maximal multiplicative persistence of $k$-cycles in random Čech complexes. Let $f:\\mathbb{R}^d\\to\\mathbb{R}$ be a probability density function, let ${P}_n$ be a Poisson process with intensity $nf$, and let $Π_{k,n}$ denote the largest death-to-birth ratio among all non-essential $k$-cycles ($1\\le k \\le d-1$). For the uniform distribution in the unit hypercube, it was proved in [9] that $Π_{k,n} = Θ\\left( \\left(\\frac{\\log n}{\\log\\log n}\\right)^{1/k}\\right)$. In this paper we sharpen and extend this result to a law of large numbers, for a broad class of distributions. Most significantly, we show that the limiting constant depends only on $d$ and $k$, and not on the probability density $f$. Thus the extreme value of multiplicative persistence exhibits a universality phenomenon. We show that the limiting constant is determined by the asymptotic covering density of the $k$-dimensional sphere. Our proof identifies the geometric mechanism underlying maximal cycles, a persistent isoperimetric inequality, which gives sharp bounds on the number of points needed to generate a highly persistent cycle. By combining covering-density estimates with isoperimetric inequalities, we show that this minimum is asymptotically attained by efficient coverings of a $k$-sphere. A key ingredient is a geometric measure theory argument that uses compactness to relate discrete covering counts to the volume of a limiting cycle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19474",
  "author_total": 2,
  "authors": [
    {
      "name": "Omer Bobrowski"
    },
    {
      "name": "Primoz Skraba"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fzjeyawywcb8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19474",
  "fulltext_url": null,
  "id": "0fzjeyawywcb8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19474",
  "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.AT"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7213602256",
  "pdf_url": "https://arxiv.org/pdf/2609.19474",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fzjeyawywcb8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2609.19474",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19474"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes",
  "title_text": "A Universal Law of Large Numbers for Extreme Cycles in Random Čech Complexes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/0fzjeyawywcb8",
  "venue": "arXiv",
  "versions": []
}