{
  "abstract": "For a centered, variance-one random variable $X$ with finite fourth moment, let $x$ be the vector of square-free degree-$d$ monomials in $n$ independent copies of $X$. At the critical scale $d^2/n \\to λ\\in (0,\\infty)$, the normalized squared length of $x$ converges to the lognormal variable $R = \\exp(\\sqrt{λv}\\, Z - λv/2)$, where $v = \\mathbb{E} X^4 - 1$ and $Z$ is standard normal. If $\\binom{n}{d}/N \\to c \\in (0,\\infty)$, the sample covariance of $N$ independent copies of $x$ has an almost-sure limiting spectral law: the free compound-Poisson law with rate $1/c$ and jump distribution $\\operatorname{Law}(cR)$. It reduces to Marchenko-Pastur when $λv = 0$. The proof shows that subtracting the contribution of the sample length leaves vanishing quadratic-form fluctuations, even when $\\mathbb{E} X^3 \\neq 0$; length and direction need not be independent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.18293",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiaohui Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/612snx5q6ym70.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.18293",
  "fulltext_url": null,
  "id": "612snx5q6ym70",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.18293",
  "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"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7203858610",
  "pdf_url": "https://arxiv.org/pdf/2608.18293",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/612snx5q6ym70.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.18293",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.18293"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Critical tensor covariance at the Marchenko-Pastur threshold",
  "title_text": "Critical tensor covariance at the Marchenko-Pastur threshold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:07Z",
  "url": "https://research.draconitic.com/paper/612snx5q6ym70",
  "venue": "arXiv",
  "versions": []
}