{
  "id": "2009.13726",
  "version": "v1",
  "published": "2020-09-29T02:05:06.000Z",
  "updated": "2020-09-29T02:05:06.000Z",
  "title": "Singularity of Bernoulli matrices in the sparse regime $pn = O(\\log(n))$",
  "authors": [
    "Han Huang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider an $n\\times n$ random matrix $A_n$ with i.i.d Bernoulli($p$) entries. In a recent result of Litvak-Tikhomirov, they proved the conjecture $$ \\mathbb{P}\\{\\mbox{$A_n$ is singular}\\}=(1+o_n(1)) \\mathbb{P}\\big\\{\\mbox{either a row or a column of $A_n$ equals zero}\\big\\}. $$ for $ C\\frac{\\log(n)}{n} \\le p \\le \\frac{1}{C}$ for some large constant $C>1$. In this paper, we setted this conjecture in the sparse regime when $p$ satisfies $$ 1 \\le \\liminf_{n\\rightarrow \\infty} \\frac{pn}{\\log(n)} \\le \\limsup_{n\\rightarrow \\infty} \\frac{pn}{\\log(n)} < + \\infty. $$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-29T02:05:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sparse regime",
      "bernoulli matrices",
      "singularity",
      "random matrix",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}