{
  "id": "1908.01653",
  "version": "v1",
  "published": "2019-08-05T14:48:27.000Z",
  "updated": "2019-08-05T14:48:27.000Z",
  "title": "Optimal Lower Bound on the Least Singular Value of the Shifted Ginibre Ensemble",
  "authors": [
    "Giorgio Cipolloni",
    "László Erdős",
    "Dominik Schröder"
  ],
  "comment": "33 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the least singular value of a large random matrix with real or complex i.i.d. Gaussian entries shifted by a constant $z\\in\\mathbb{C}$. We prove an optimal lower tail estimate on this singular value in the critical regime where $z$ is around the spectral edge thus improving the classical bound of [Sankar, Spielman, Teng, 2006] in the edge regime. Lacking Br\\'ezin-Hikami formulas in the real case, we rely on the superbosonization formula [Littelmann, Sommers, Zirnbauer, 2008].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-05T14:48:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52"
    ],
    "keywords": [
      "singular value",
      "optimal lower bound",
      "shifted ginibre ensemble",
      "optimal lower tail estimate",
      "large random matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}