{
  "id": "1507.06520",
  "version": "v1",
  "published": "2015-07-23T14:51:21.000Z",
  "updated": "2015-07-23T14:51:21.000Z",
  "title": "Quantum ergodicity for quantum graphs without back-scattering",
  "authors": [
    "Matthew Brammall",
    "Brian Winn"
  ],
  "comment": "28 pages, 5 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We give an estimate of the quantum variance for $d$-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random $d$-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-23T14:51:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q35",
      "34L20"
    ],
    "keywords": [
      "quantum graphs",
      "regular graphs",
      "back-scattering",
      "quantum ergodicity holds",
      "boundary scattering matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150706520B"
    }
  }
}