{
  "id": "2105.13974",
  "version": "v1",
  "published": "2021-05-28T16:53:41.000Z",
  "updated": "2021-05-28T16:53:41.000Z",
  "title": "Level-set percolation of the Gaussian free field on regular graphs III: giant component on expanders",
  "authors": [
    "Jiří Černý"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\\ge 3$. This class includes $d$-regular expanders of large girth and typical realisations of random $d$-regular graphs. We show that the level set of the zero-average Gaussian free field above level $h$ has a giant component in the whole supercritical phase, that is for all $h<h_\\star$, with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [AC20b], where it was shown that a linear fraction of vertices is in mesoscopic components if $h<h_\\star$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-28T16:53:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60G15",
      "05C48"
    ],
    "keywords": [
      "regular graphs",
      "giant component",
      "level-set percolation",
      "zero-average gaussian free field",
      "mesoscopic components"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}