{
  "abstract": "Let $M_n$ be an $n\\times n$ random matrix with entries in $\\{-1, 0, 1\\}$, where each row is independently and uniformly sampled from the set of all vectors in $\\{-1, 0, 1\\}^n$ containing exactly $d$ nonzero entries, with $d=pn$ for some fixed constant $p\\in(0,1/2]$. A recent result by one of the authors gives a lower bound of order $n^{-1/2}$ for the smallest singular value $s_n(M_n)$ of this model with high probability. In this paper, we establish a matching upper bound for $s_n(M_n)$. Specifically, there exists a constant $C_p>0$ depending only on $p$ such that \\[ \\forall \\varepsilon >0 \\qquad \\mathbb{P} \\left( s_n(M_n) \\le \\frac{\\sqrt{d}}{\\varepsilon^2 n} \\right) \\ge 1-C_p\\left(\\varepsilon + \\frac{1}{\\sqrt{d}}\\right). \\] Consequently, in the dense regime, the smallest singular value $s_n(M_n)$ of signed random combinatorial matrices is typically of order $n^{-1/2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.33427",
  "author_total": 4,
  "authors": [
    {
      "name": "Yuxiao Shi"
    },
    {
      "name": "Zeyan Song"
    },
    {
      "name": "Kexin Yu"
    },
    {
      "name": "Tingzhou Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9jgf19xsdk2k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.33427",
  "fulltext_url": null,
  "id": "e9jgf19xsdk2k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.33427",
  "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.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7223860835",
  "pdf_url": "https://arxiv.org/pdf/2610.33427",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9jgf19xsdk2k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.33427",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.33427"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An upper bound on the smallest singular value of dense signed random combinatorial matrices",
  "title_text": "An upper bound on the smallest singular value of dense signed random combinatorial matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/e9jgf19xsdk2k",
  "venue": "arXiv",
  "versions": []
}