{
  "id": "1811.04700",
  "version": "v1",
  "published": "2018-11-12T12:58:27.000Z",
  "updated": "2018-11-12T12:58:27.000Z",
  "title": "The random walk penalised by its range in dimensions $d\\geq 3$",
  "authors": [
    "Nathanael Berestycki",
    "Raphael Cerf"
  ],
  "comment": "70 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a self-attractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $\\exp ( - |R_N|)$, where $R_N$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately $\\rho_d N^{1/(d+2)}$, for some explicit constant $\\rho_d >0$. This proves a conjecture of Bolthausen who obtained this result in the case $d=2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-12T12:58:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dimensions",
      "solid euclidean ball",
      "self-attractive random walk",
      "explicit constant",
      "factor proportional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}