{
  "id": "2209.11732",
  "version": "v1",
  "published": "2022-09-23T17:25:16.000Z",
  "updated": "2022-09-23T17:25:16.000Z",
  "title": "Replica approach to the generalized Rosenzweig-Porter model",
  "authors": [
    "Davide Venturelli",
    "Leticia F. Cugliandolo",
    "Grégory Schehr",
    "Marco Tarzia"
  ],
  "comment": "Submission to SciPost; 44 pages, 7 figures",
  "categories": [
    "cond-mat.dis-nn",
    "cond-mat.stat-mech"
  ],
  "abstract": "The generalized Rosenzweig--Porter model arguably constitutes the simplest random matrix ensemble displaying a non-ergodic delocalized phase, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. We confirm our results with numerical tests.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-23T17:25:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "replica approach",
      "random matrix ensemble displaying",
      "level compressibility",
      "simplest random matrix",
      "average spectral density"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}