{
  "abstract": "We study the classical $XY$-model on random geometric graphs $\\mathcal{G}_{n, \\varepsilon}$, which are obtained by sampling $n \\in \\mathbb{N}$ independent points in a finite domain $Ω\\subset \\mathbb{R}^d$, $d \\geq 2$, and connecting two points by and edge if their distance is of order $\\varepsilon > 0$. We refer to $\\mathcal{G}_{n, \\varepsilon}$ as the random environment. Letting $\\varepsilon \\to 0$ as $n \\to \\infty$ at a sufficiently slow rate, these graphs capture the geometry of $Ω$. Denoting the inverse temperature by $β$, we show that in the limit $β\\to \\infty$ at a rate depending on $n$ and $\\varepsilon$, the $XY$-model on $\\mathcal{G}_{n, \\varepsilon}$ exhibits long range order in the sense that we prove a lower bound away from zero on the two-point function. Our result is quenched in the random environment: long-range order holds with large probability, converging to one as $n \\to \\infty$. To prove the statement, we show that with high probability the environment is sufficiently regular to apply a convexity argument and the Brascamp--Lieb inequality.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02618",
  "author_total": 2,
  "authors": [
    {
      "name": "Margherita Disertori"
    },
    {
      "name": "Max Mihailescu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftd33tzwarhma.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02618",
  "fulltext_url": null,
  "id": "ftd33tzwarhma",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02618",
  "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": "W7207743480",
  "pdf_url": "https://arxiv.org/pdf/2609.02618",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftd33tzwarhma.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:16Z",
      "native_id": "2609.02618",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02618"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Asymptotic long-range order for the XY-model on random geometric graphs",
  "title_text": "Asymptotic long-range order for the XY-model on random geometric graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/ftd33tzwarhma",
  "venue": "arXiv",
  "versions": []
}