{
  "id": "1112.2016",
  "version": "v4",
  "published": "2011-12-09T04:36:00.000Z",
  "updated": "2012-11-09T19:05:05.000Z",
  "title": "Local semicircle law in the bulk for Gaussian $β$-ensemble",
  "authors": [
    "Philippe Sosoe",
    "Percy Wong"
  ],
  "comment": "31 pages, version 3 corrected typos and expanded sections 4-5",
  "journal": "J. Stat. Phys. 148 (2) 204-232 (2012)",
  "categories": [
    "math.PR"
  ],
  "abstract": "We use the tridiagonal matrix representation to derive a local semicircle law for Gaussian beta ensembles at the optimal level of $n^{-1+\\delta}$ for any $\\delta > 0$. Using a resolvent expansion, we first derive a semicircle law at the intermediate level of $n^{-1/2+\\delta}$; then an induction argument allows us to reach the optimal level. This result was obtained in a different setting, using different methods, by Bourgade, Erd\\\"os, and Yau and in Bao and Su. Our approach is new and could be extended to other tridiagonal models.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-11-09T19:05:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "60K35"
    ],
    "keywords": [
      "local semicircle law",
      "optimal level",
      "gaussian beta ensembles",
      "tridiagonal matrix representation",
      "resolvent expansion"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s10955-012-0536-4",
      "journal": "Journal of Statistical Physics",
      "year": 2012,
      "month": "Aug",
      "volume": 148,
      "number": 2,
      "pages": 204
    },
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012JSP...148..204S"
    }
  }
}