{
  "abstract": "Let $p\\in(0,1/2]$ be fixed, and let $d,n$ be positive integers satisfying $d=pn$. Let $Q_n$ be an $n\\times n$ random combinatorial matrix whose rows are independent and uniformly distributed over the vectors in $\\{0,1\\}^n$ having exactly $d$ ones. For an arbitrary deterministic invertible $n\\times n$ matrix $M$, we establish two-sided bounds for the smallest singular value of $Q_nM$. More precisely, for every $\\varepsilon\\in(0,1)$, \\[ \\mathbb P\\!\\left( \\varepsilon\\le s_n(Q_nM)\\cdot \\|M^{-1}\\|_{\\mathrm{HS}} \\le \\varepsilon^{-2} \\right) \\ge 1-C\\left( \\varepsilon+d^{-1/2}+e^{-c\\,\\operatorname{sr}(M^{-1})} \\right)-2e^{-cn}, \\] where positive constants $C, c$ depend only on $p$ and $\\operatorname{sr}$ denotes the stable rank. Consequently, for any sequence of deterministic invertible matrices $M$ satisfying $\\operatorname{sr}(M^{-1})\\to\\infty$, $s_n(Q_nM)$ is typically of order $\\|M^{-1}\\|_{\\mathrm{HS}}^{-1}$. In particular, taking $M=\\mathbf I_n$ recovers the known $n^{-1/2}$ scale for $s_n(Q_n)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38588",
  "author_total": 3,
  "authors": [
    {
      "name": "Yuxiao Shi"
    },
    {
      "name": "Kexin Yu"
    },
    {
      "name": "Tingzhou Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hsw1wbmtstvy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38588",
  "fulltext_url": null,
  "id": "0hsw1wbmtstvy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38588",
  "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/2610.38588",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hsw1wbmtstvy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38588",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38588"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The smallest singular value of products of random combinatorial and deterministic matrices",
  "title_text": "The smallest singular value of products of random combinatorial and deterministic matrices",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/0hsw1wbmtstvy",
  "venue": "arXiv",
  "versions": []
}