{
  "id": "2109.03227",
  "version": "v1",
  "published": "2021-09-07T17:55:13.000Z",
  "updated": "2021-09-07T17:55:13.000Z",
  "title": "The completely delocalized region of the Erdős-Rényi graph",
  "authors": [
    "Johannes Alt",
    "Raphael Ducatez",
    "Antti Knowles"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We analyse the eigenvectors of the adjacency matrix of the Erd\\H{o}s-R\\'enyi graph on $N$ vertices with edge probability $\\frac{d}{N}$. We determine the full region of delocalization by determining the critical values of $\\frac{d}{\\log N}$ down to which delocalization persists: for $\\frac{d}{\\log N} > \\frac{1}{\\log 4 - 1}$ all eigenvectors are completely delocalized, and for $\\frac{d}{\\log N} > 1$ all eigenvectors with eigenvalues away from the spectral edges are completely delocalized. Below these critical values, it is known [arXiv:2005.14180, arXiv:2106.12519] that localized eigenvectors exist in the corresponding spectral regions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-07T17:55:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52",
      "05C80"
    ],
    "keywords": [
      "erdős-rényi graph",
      "delocalized region",
      "eigenvectors",
      "critical values",
      "spectral edges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}