{
  "abstract": "We study sample covariance matrices $K = \\frac{1}{N} \\sum_{i=1}^N \\mathbf{x}_i \\mathbf{x}_i^* \\in \\mathbb{R}^{n \\times n}$ in the proportional regime $n \\asymp N$. The columns $ \\mathbf{x}_1, \\ldots, \\mathbf{x}_N \\in \\mathbb{R}^n$ are independent and centered, with common covariance $\\mathbb{E} \\mathbf{x}_i \\mathbf{x}_i^* = Σ$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \\mathbf{x}_i^* A \\mathbf{x}_i - \\mathrm{Tr} ΣA | \\prec \\| A \\|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $Σ$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $η:= \\mathrm{Im}\\, z \\geq N^{-1 + τ}$, \\[ \\big| \\langle \\mathbf{u} , \\big( (K-z)^{-1} - (-zI_n-z\\widetilde m_0(z)Σ\\big)^{-1} \\big) \\mathbf{v} \\> \\big| \\prec \\sqrt{\\frac{\\mathrm{Im}\\, \\widetilde m_0 (z)}{Nη}} + \\frac{1}{Nη} \\] for all deterministic unit vectors $ \\mathbf{u}, \\mathbf{v} \\in \\mathbb{C}^n$, where $\\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09440",
  "author_total": 2,
  "authors": [
    {
      "name": "Renyuan Ma"
    },
    {
      "name": "Theodor Misiakiewicz"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26dztqqjjp37c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09440",
  "fulltext_url": null,
  "id": "26dztqqjjp37c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09440",
  "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": [
    [
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      "math-ph"
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      "openalex",
      "2613|Random Matrices and Applications"
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  "openalex": "W7212165695",
  "pdf_url": "https://arxiv.org/pdf/2609.09440",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26dztqqjjp37c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:07Z",
      "native_id": "2609.09440",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09440"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "The anisotropic local law for sample covariance matrices under quadratic-form concentration",
  "title_text": "The anisotropic local law for sample covariance matrices under quadratic-form concentration",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:17Z",
  "url": "https://research.draconitic.com/paper/26dztqqjjp37c",
  "venue": "arXiv",
  "versions": []
}