{
  "id": "1912.08790",
  "version": "v1",
  "published": "2019-12-18T18:42:11.000Z",
  "updated": "2019-12-18T18:42:11.000Z",
  "title": "Recurrence of the Uniform Infinite Half-Plane Map via duality of resistances",
  "authors": [
    "Thomas Budzinski",
    "Thomas Lehéricy"
  ],
  "comment": "34 pages, 10 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the simple random walk on the Uniform Infinite Half-Plane Map, which is the local limit of critical Boltzmann planar maps with a large and simple boundary. We prove that the simple random walk is recurrent, and that the resistance between the root and the boundary of the hull of radius $r$ is at least of order $\\log r$. This resistance bound is expected to be sharp, and is better than those following from previous proofs of recurrence for non bounded-degree planar maps models. Our main tools are the self-duality of uniform planar maps, a classical lemma about duality of resistances and some peeling estimates. The proof shares some ideas with Russo--Seymour--Welsh theory in percolation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-18T18:42:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniform infinite half-plane map",
      "resistance",
      "simple random walk",
      "non bounded-degree planar maps models",
      "recurrence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}