{
  "id": "2104.11898",
  "version": "v1",
  "published": "2021-04-24T07:34:46.000Z",
  "updated": "2021-04-24T07:34:46.000Z",
  "title": "Capacity of the range of branching random walks in low dimensions",
  "authors": [
    "Tianyi Bai",
    "Yueyun Hu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a branching random walk $(V_u)_{u\\in \\mathcal T^{IGW}}$ in $\\mathbb Z^d$ with the genealogy tree $\\mathcal T^{IGW}$ formed by a sequence of i.i.d. critical Galton-Watson trees. Let $R_n $ be the set of points in $\\mathbb Z^d$ visited by $(V_u)$ when the index $u$ explores the first $n$ subtrees in $\\mathcal T^{IGW}$. Our main result states that for $d\\in \\{3, 4, 5\\}$, the capacity of $R_n$ is almost surely equal to $n^{\\frac{d-2}{2}+o(1)}$ as $n \\to \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-24T07:34:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60J65"
    ],
    "keywords": [
      "branching random walk",
      "low dimensions",
      "main result states",
      "genealogy tree",
      "critical galton-watson trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}