{
  "abstract": "Let $A=(ξ_{ij})$ be an $n\\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \\[ \\mathbb Eξ_{ij}=0,\\qquad \\mathbb Eξ_{ij}^{2}=1,\\qquad \\sup_{z\\in\\mathbb R}\\mathbb P(|ξ_{ij}-z|<a)\\le b \\] for fixed $a>0$ and $b\\in(0,1)$. We prove that, for every $δ\\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,δ$, such that \\[ \\mathbb P\\left( s_{n+1-l}(A)>Ct\\frac l{\\sqrt n} \\right) \\le \\exp\\!\\left(-c\\min\\{tl,n\\}\\right) \\] for every $t\\ge1$ and every $1\\le l\\le(1-δ)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\\asymp l/\\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\\sqrt{N+1}-\\sqrt{n-l+1}$ for $N\\times n$ matrices whenever $N-n+l\\le(1-δ)N$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30722",
  "author_total": 2,
  "authors": [
    {
      "name": "Achintya Raya Polavarapu"
    },
    {
      "name": "Manuel Fernandez V"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bdqz2rz5cvkbh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30722",
  "fulltext_url": null,
  "id": "bdqz2rz5cvkbh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30722",
  "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": "W7204955754",
  "pdf_url": "https://arxiv.org/pdf/2608.30722",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bdqz2rz5cvkbh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30722",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30722"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Intermediate Singular Values of Random Matrices under Second-Moment and Anti-Concentration Assumptions",
  "title_text": "Intermediate Singular Values of Random Matrices under Second-Moment and Anti-Concentration Assumptions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/bdqz2rz5cvkbh",
  "venue": "arXiv",
  "versions": []
}