{
  "abstract": "In this work we study the maximal ratio $I$ between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball $B_L$. Our main result establishes the leading-order asymptotics $\\mathbb{E}I\\approx i\\ln^{3/4}L$ for some $i\\in(0,\\infty)$ and superconcentration at scale $O(\\ln^{-1/4}L)$. This is done by relating to a simpler $(1+1)$-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08726",
  "author_total": 3,
  "authors": [
    {
      "name": "Xiaopeng Cheng"
    },
    {
      "name": "Felix Otto"
    },
    {
      "name": "Matteo Palmieri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/348ce080bfaq1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08726",
  "fulltext_url": null,
  "id": "348ce080bfaq1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08726",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7211992167",
  "pdf_url": "https://arxiv.org/pdf/2609.08726",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/348ce080bfaq1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:52Z",
      "native_id": "2609.08726",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08726"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Asymptotics of a planar isoperimetric problem with a white-noise volume term",
  "title_text": "Asymptotics of a planar isoperimetric problem with a white-noise volume term",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/348ce080bfaq1",
  "venue": "arXiv",
  "versions": []
}