{
  "id": "2401.12935",
  "version": "v1",
  "published": "2024-01-23T17:43:12.000Z",
  "updated": "2024-01-23T17:43:12.000Z",
  "title": "The local limit of rooted directed animals on the square lattice",
  "authors": [
    "Olivier Hénard",
    "Édouard Maurel-Segala",
    "Arvind Singh"
  ],
  "comment": "59 pages, 16 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We consider the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed by letting gravity act on a right-continuous random walk and one as a Markov process, obtained by slicing the animal horizontally. We look at geometric properties of this local limit and prove, in particular, that it consists of a single vertex at infinitely many (random) levels. Several martingales are found in connection with the confinement of the infinite directed animal on the non-negative coordinates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-23T17:43:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B41",
      "60K35"
    ],
    "keywords": [
      "local limit",
      "rooted directed animals",
      "square lattice",
      "uniformly distributed directed animals",
      "resulting uniform infinite directed animal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 59,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}