{
  "id": "1403.2582",
  "version": "v2",
  "published": "2014-03-11T14:30:03.000Z",
  "updated": "2014-11-17T17:58:52.000Z",
  "title": "Finite size correction to the spectrum of regular random graphs: an analytical solution",
  "authors": [
    "Fernando L. Metz",
    "Giorgio Parisi",
    "Luca Leuzzi"
  ],
  "comment": "Extended version with an extra appendix explaining the connection with rigorous results",
  "journal": "Phys. Rev. E 90, 052109 (2014)",
  "doi": "10.1103/PhysRevE.90.052109",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "We develop a thorough analytical study of the $O(1/N)$ correction to the spectrum of regular random graphs with $N \\rightarrow \\infty$ nodes. The finite size fluctuations of the resolvent are given in terms of a weighted series over the contributions coming from loops of all possible lengths, from which we obtain the isolated eigenvalue as well as an analytical expression for the $O(1/N)$ correction to the continuous part of the spectrum. The comparison between this analytical formula and direct diagonalization results exhibits an excellent agreement, confirming the correctness of our expression.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-11T14:30:03.000Z",
      "comment": "17 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-17T17:58:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.-a",
      "89.75.Hc",
      "71.23.-k"
    ],
    "keywords": [
      "regular random graphs",
      "finite size correction",
      "analytical solution",
      "direct diagonalization results",
      "thorough analytical study"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review E",
      "year": 2014,
      "month": "Nov",
      "volume": 90,
      "number": 5,
      "pages": "052109"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014PhRvE..90e2109M"
    }
  }
}