{
  "id": "2203.06141",
  "version": "v1",
  "published": "2022-03-11T18:13:25.000Z",
  "updated": "2022-03-11T18:13:25.000Z",
  "title": "The least singular value of a random symmetric matrix",
  "authors": [
    "Marcelo Campos",
    "Matthew Jenssen",
    "Marcus Michelen",
    "Julian Sahasrabudhe"
  ],
  "comment": "42 pages + 30 page supplement. The supplement builds on our previous work arXiv:2105.11384 and provides the proof of a technical quasi-randomness statement",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $A$ be a $n \\times n$ symmetric matrix with $(A_{i,j})_{i\\leq j} $, independent and identically distributed according to a subgaussian distribution. We show that $$\\mathbb{P}(\\sigma_{\\min}(A) \\leq \\varepsilon/\\sqrt{n}) \\leq C \\varepsilon + e^{-cn},$$ where $\\sigma_{\\min}(A)$ denotes the least singular value of $A$ and the constants $C,c>0 $ depend only on the distribution of the entries of $A$. This result confirms a folklore conjecture on the lower-tail asymptotics of the least singular value of random symmetric matrices and is best possible up to the dependence of the constants on the distribution of $A_{i,j}$. Along the way, we prove that the probability $A$ has a repeated eigenvalue is $e^{-\\Omega(n)}$, thus confirming a conjecture of Nguyen, Tao and Vu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-11T18:13:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random symmetric matrix",
      "singular value",
      "subgaussian distribution",
      "result confirms",
      "folklore conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}