{
  "id": "2304.08647",
  "version": "v1",
  "published": "2023-04-17T22:36:27.000Z",
  "updated": "2023-04-17T22:36:27.000Z",
  "title": "Speed of the random walk on the supercritical Gaussian Free Field percolation on regular trees",
  "authors": [
    "Guillaume Conchon--Kerjan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper, we study the random walk on a supercritical branching process with an uncountable and unbounded set of types supported on the $d$-regular tree $\\mathbb{T}_d$ ($d\\geq 3$), namely the cluster $\\mathcal{C}_\\circ^h$ of the root in the level set of the Gaussian Free Field (GFF) above an arbitrary value $h\\in (-\\infty, h_{\\star})$. The value $h_{\\star}\\in (0,\\infty)$ is the percolation threshold; in particular, $\\mathcal{C}_\\circ^h$ is infinite with positive probability. We show that on $\\mathcal{C}_\\circ^h$ conditioned to be infinite, the simple random walk is ballistic, and we give a law of large numbers and a Donsker theorem for its speed. To do so, we design a renewal construction that withstands the long-range dependencies in the structure of the tree. This allows us to translate underlying ergodic properties of $\\mathcal{C}_\\circ^h$ into regularity estimates for the random walk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-17T22:36:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60G50",
      "60J80",
      "60G15"
    ],
    "keywords": [
      "supercritical gaussian free field percolation",
      "regular tree",
      "simple random walk",
      "arbitrary value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}