{
  "abstract": "We resolve the Rademacher singularity conjecture: an $n\\times n$ matrix with independent uniform $\\{-1,1\\}$ entries is singular with probability $(2+o(1))n^22^{-n}$. More generally, for an $n\\times n$ matrix $M_n$ with independent entries uniform on a fixed set $S\\subset\\mathbb{R}$ of cardinality $q\\ge2$, we prove $\\mathbb{P}(s_n(M_n)\\le z/\\sqrt{n}) \\le Cz+a_n(S)+C'n^{1+\\varepsilon}q^{-n}$ for all $n\\ge1$, $z\\ge0$, and $\\varepsilon>0$, with $C=C(S)$ and $C'=C'(S,\\varepsilon)$. Here $s_n$ denotes the least singular value, and $a_n(S)$ is the sum of the probabilities of the events involving zero rows or columns and equal or opposite pairs of rows or columns.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39463",
  "author_total": 3,
  "authors": [
    {
      "name": "Xuanang Hu"
    },
    {
      "name": "Zeyan Song"
    },
    {
      "name": "Xinglong Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dc8exavh4ywzp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39463",
  "fulltext_url": null,
  "id": "dc8exavh4ywzp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39463",
  "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": "W7215126740",
  "pdf_url": "https://arxiv.org/pdf/2609.39463",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dc8exavh4ywzp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2609.39463",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39463"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp singularity asymptotics for discrete random matrices",
  "title_text": "Sharp singularity asymptotics for discrete random matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/dc8exavh4ywzp",
  "venue": "arXiv",
  "versions": []
}