{
  "id": "2105.05219",
  "version": "v2",
  "published": "2021-05-11T17:35:18.000Z",
  "updated": "2021-06-13T20:47:58.000Z",
  "title": "Sharp phase transition for Gaussian percolation in all dimensions",
  "authors": [
    "Franco Severo"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the level-sets of continuous Gaussian fields on $\\mathbb{R}^d$ above a certain level $-\\ell\\in \\mathbb{R}$, which defines a percolation model as $\\ell$ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than $d$ (in particular, this includes the Bargmann-Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point $\\ell_c$. More precisely, we show that connection probabilities decay exponentially for $\\ell<\\ell_c$ and percolation occurs in sufficiently thick 2D slabs for $\\ell>\\ell_c$. This extends results recently obtained in dimension $d=2$ to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice $\\varepsilon\\mathbb{Z}^d$) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter $\\ell$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-06-13T20:47:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B43",
      "60K35",
      "60G15",
      "60G60"
    ],
    "keywords": [
      "sharp phase transition",
      "gaussian percolation",
      "connection probabilities decay",
      "covariance kernel satisfies",
      "sufficiently thick 2d slabs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}