{
  "abstract": "We study the smallest nonzero eigenvalue of the sample correlation matrix $\\mathbf{R}_n$ formed from a $p_n \\times n$ data matrix with i.i.d. real entries $ξ$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \\to ϕ\\in (0, \\infty) \\setminus \\{1\\}$. In the tall regime $ϕ> 1$, we prove almost-sure convergence of $λ_n (\\mathbf{R}_n)$ to the lower Marčenko--Pastur edge $λ_- = (1 - \\sqrtϕ)^2$ without additional moment assumptions. In the wide regime $ϕ< 1$, we establish a phase transition at the third-order tail scale. If $t^3 \\mathbb{P}\\{\\lvert ξ\\rvert > t\\} \\to 0$ as $t \\to \\infty$, the smallest eigenvalue $λ_{p_n} (\\mathbf{R}_n)$ converges in probability to $λ_-$. While if $t^3 \\mathbb{P}\\{\\lvert ξ\\rvert > t\\} \\to \\infty$, then $λ_{p_n} (\\mathbf{R}_n)$ converges in probability to zero. At the critical scale $t^3 \\mathbb{P}\\{\\lvert ξ\\rvert > t\\} \\to κ\\in (0, \\infty)$, the point process of eigenvalues in the lower gap $(0, λ_-)$ converges in distribution to a Poisson random measure with explicit intensity. In this critical regime, we also identify the nondegenerate limiting distribution of $λ_{p_n} (\\mathbf{R}_n)$, which has a continuous density on $(0, λ_-)$ and a positive atom at $λ_-$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15731",
  "author_total": 4,
  "authors": [
    {
      "name": "Zeqin Lin"
    },
    {
      "name": "Guamgming Pan"
    },
    {
      "name": "Haozhu Zhao"
    },
    {
      "name": "Wang Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jj9rq5zj5t9g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15731",
  "fulltext_url": null,
  "id": "6jj9rq5zj5t9g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15731",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2609.15731",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jj9rq5zj5t9g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15731",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15731"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices",
  "title_text": "Phase transition for the smallest eigenvalue of high-dimensional sample correlation matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/6jj9rq5zj5t9g",
  "venue": "arXiv",
  "versions": []
}