{
  "id": "1909.01972",
  "version": "v1",
  "published": "2019-09-04T17:55:34.000Z",
  "updated": "2019-09-04T17:55:34.000Z",
  "title": "Level-set percolation of the Gaussian free field on regular graphs II: Finite expanders",
  "authors": [
    "Angelo Abächerli",
    "Jiří Černý"
  ],
  "comment": "42 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the zero-average Gaussian free field on a certain class of finite $d$-regular graphs for fixed $d\\geq 3$. 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 $h$ exhibits a phase transition at level $h_\\star$, which agrees with the critical value for level-set percolation of the Gaussian free field on the infinite $d$-regular tree. More precisely, we show that, with probability tending to one as the size of the finite graphs tends to infinity, the level set above level $h$ does not contain any connected component of larger than logarithmic size whenever $h>h_\\star$, and on the contrary, whenever $h<h_\\star$, a linear fraction of the vertices is contained in connected components of the level set above level $h$ having a size of at least a small fractional power of the total size of the graph. It remains open whether in the supercritical phase $h<h_\\star$, as the size of the graphs tends to infinity, one observes the emergence of a (potentially unique) giant connected component of the level set above level $h$. The proofs in this article make use of results from the accompanying paper [AC1].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-04T17:55:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regular graphs",
      "level-set percolation",
      "finite expanders",
      "level set",
      "zero-average gaussian free field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}