{
  "abstract": "Let $A$ be an $n\\times n$ real symmetric random matrix whose entries above the diagonal are independent, each with largest atom at most $β\\in (0,1)$. The diagonal entries have an arbitrary joint distribution and are independent of the entries above the diagonal. We prove that \\[ \\mathbb{P}[\\mathrm{rank}(A)\\le n-k]\\le 2e^{-c_βnk},\\qquad 1\\le k\\le n, \\] where $c_β>0$ depends only on $β$. This extends Han's corank bound to the full range $1\\le k\\le n$. No moment assumptions are needed, and the entries need not be centered or identically distributed. In particular, the case $k=1$ gives an exponential singularity bound under the uniform atom assumption. Our argument also gives a new proof of the exponential singularity bound for symmetric Rademacher matrices obtained by Campos, Jenssen, Michelen, and Sahasrabudhe. For every fixed $p\\in (0,1)$, it yields the exponential singularity bound conjectured by Costello and Vu for the adjacency matrix of the Erdős--Rényi random graph $G(n,p)$, where the case $p=1/2$ was previously proved by Han.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39424",
  "author_total": 3,
  "authors": [
    {
      "name": "Huaijin Liang"
    },
    {
      "name": "Kexin Yu"
    },
    {
      "name": "Tingzhou Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d72d3v42j9166.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39424",
  "fulltext_url": null,
  "id": "d72d3v42j9166",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39424",
  "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/2610.39424",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d72d3v42j9166.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39424",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39424"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Exponential singularity and rank bounds for random symmetric matrices",
  "title_text": "Exponential singularity and rank bounds for random symmetric matrices",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/d72d3v42j9166",
  "venue": "arXiv",
  "versions": []
}