{
  "abstract": "We establish staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices formed from a $p_n \\times n$ data matrix with i.i.d. real entries of mean zero and unit variance, allowing an infinite fourth moment. In the proportional regime $p_n / n \\to ϕ\\in (0, \\infty)$, the first-order asymptotics depend jointly on the aspect ratio and the entry tail. The transitions are driven by collisions of large entries in distinct rows of a common column. The first collision order capable of producing a separated upper outlier is $k_* (ϕ) = \\lfloor \\sqrtϕ \\rfloor + 2$, yielding the critical tail exponent $α_* (ϕ) = 2 + 2 / k_* (ϕ)$. This exponent decreases in steps as $ϕ$ increases, creating a staircase boundary between convergence to the upper Marčenko--Pastur edge and successive outlier levels. At exact critical tail scales, the point process of eigenvalues above the upper edge or the preceding deterministic level converges to a Poisson point process. The resulting nondegenerate limiting laws for the largest eigenvalue connect adjacent phases and have a positive atom at this baseline. If every fixed collision order is supercritical, the largest eigenvalue diverges in probability despite finite entry variance.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06731",
  "author_total": 5,
  "authors": [
    {
      "name": "Yanpeng Li"
    },
    {
      "name": "Zeqin Lin"
    },
    {
      "name": "Yiming Liu"
    },
    {
      "name": "Jiahui Xie"
    },
    {
      "name": "Haozhu Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eh30a9yzmp8tx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06731",
  "fulltext_url": null,
  "id": "eh30a9yzmp8tx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06731",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7220687759",
  "pdf_url": "https://arxiv.org/pdf/2610.06731",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eh30a9yzmp8tx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.06731",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06731"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices",
  "title_text": "Staircase phase transitions for the largest eigenvalue of heavy-tailed sample correlation matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/eh30a9yzmp8tx",
  "venue": "arXiv",
  "versions": []
}