{
  "abstract": "The set of $n\\times n$ correlation matrices, known as the elliptope, has volume decaying at the super-exponential rate $\\exp\\{-\\tfrac14 n^2\\log n\\}$. We characterize where this vanishing volume concentrates. A uniform draw is entrywise close to the identity yet globally far from it and nearly singular. Its maximum absolute correlation is of order $\\sqrt{\\log n/n}$, its Frobenius distance is asymptotic to $\\sqrt n$, its empirical spectral distribution converges to the Marchenko-Pastur law with ratio one, and its smallest eigenvalue has the exact $\\operatorname{Beta}(1,d)$ distribution, where $d=n(n-1)/2$, and is therefore of order $n^{-2}$. Distinct off-diagonal entries are exactly pairwise independent under every $\\operatorname{LKJ}(η)$ law, which yields a Chen-Stein proof of the extreme-correlation point-process limit for the whole LKJ family and an $O(n^{-1})$ total-variation bound for finite-dimensional exceedance counts relative to Poisson laws with their exact finite-$n$ means. Finally, for a bounded, centered i.i.d. off-diagonal perturbation of any correlation matrix, the nearest-correlation projection removes a fraction of the squared perturbation that tends to one and recovers the original matrix in average squared error per entry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.04162",
  "author_total": 1,
  "authors": [
    {
      "name": "Peter Reinhard Hansen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6ytcmne7hzgxm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.04162",
  "fulltext_url": null,
  "id": "6ytcmne7hzgxm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.04162",
  "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",
      "econ.EM"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7196987908",
  "pdf_url": "https://arxiv.org/pdf/2608.04162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6ytcmne7hzgxm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2608.04162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.04162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes",
  "title_text": "Correlation Matrices in High Dimensions: Volume, Spectrum, and Extremes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/6ytcmne7hzgxm",
  "venue": "arXiv",
  "versions": []
}