{
  "id": "2112.09116",
  "version": "v2",
  "published": "2021-12-16T18:56:49.000Z",
  "updated": "2023-01-23T15:53:57.000Z",
  "title": "Phase transition for level-set percolation of the membrane model in dimensions $d \\geq 5$",
  "authors": [
    "Alberto Chiarini",
    "Maximilian Nitzschner"
  ],
  "comment": "29 pages, 1 figure, to appear in Journal of Statistical Physics",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider level-set percolation for the Gaussian membrane model on $\\mathbb{Z}^d$, with $d \\geq 5$, and establish that as $h \\in \\mathbb{R}$ varies, a non-trivial percolation phase transition for the level-set above level $h$ occurs at some finite critical level $h_\\ast$, which we show to be positive in high dimensions. Along $h_\\ast$, two further natural critical levels $h_{\\ast\\ast}$ and $\\overline{h}$ are introduced, and we establish that $-\\infty <\\overline{h} \\leq h_\\ast \\leq h_{\\ast\\ast} < \\infty$, in all dimensions. For $h > h_{\\ast\\ast}$, we find that the connectivity function of the level-set above $h$ admits stretched exponential decay, whereas for $h < \\overline{h}$, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, R\\'ath and Sapozhnikov, see arXiv:1212.2885, for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-01-23T15:53:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60K35",
      "60G60",
      "82B43"
    ],
    "keywords": [
      "level-set percolation",
      "dimensions",
      "non-trivial percolation phase transition",
      "admits stretched exponential decay",
      "general correlated percolation models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}