{
  "abstract": "For even $n$, let $P_n$ be the probability that independent uniform strict preference lists on $n$ participants admit a stable perfect matching. We prove \\[ P_n\\sim\\frac{e\\,2^{1/4}Γ(3/4)}{\\sqrtπ}\\,n^{-1/4}. \\] This establishes Mertens's conjectured exponent of decay, with a leading constant different from his original numerical prediction. The proof starts from Mertens's exact alternating sum over stable permutations. To preserve its cancellation, we construct a common approximation for all cycle structures with the same number of vertices in cycles longer than two. By symmetry, the integrated first-order correction is the same for every such cycle structure, and the remaining errors can be summed in absolute value. The enumeration then reduces the probability to a one-dimensional sum with positive terms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07741",
  "author_total": 2,
  "authors": [
    {
      "name": "Caden Young"
    },
    {
      "name": "Ian Studebaker"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ab2ey21hhgj32.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07741",
  "fulltext_url": null,
  "id": "ab2ey21hhgj32",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07741",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2002|Game Theory and Voting Systems"
    ]
  ],
  "openalex": "W7220929396",
  "pdf_url": "https://arxiv.org/pdf/2610.07741",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ab2ey21hhgj32.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07741",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07741"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Asymptotics for the Solvability Probability of Random Stable Roommates",
  "title_text": "Sharp Asymptotics for the Solvability Probability of Random Stable Roommates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/ab2ey21hhgj32",
  "venue": "arXiv",
  "versions": []
}