{
  "id": "1405.1583",
  "version": "v1",
  "published": "2014-05-07T12:18:31.000Z",
  "updated": "2014-05-07T12:18:31.000Z",
  "title": "The harmonic measure of balls in critical Galton-Watson trees with infinite variance offspring distribution",
  "authors": [
    "Shen Lin"
  ],
  "comment": "46 pages, 4 figures. arXiv admin note: substantial text overlap with arXiv:1304.7190 by other authors",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study properties of the harmonic measure of balls in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $\\alpha\\in (1,2]$. Here the harmonic measure refers to the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is roughly of order $n^{\\frac{1}{\\alpha-1}}$, most of the harmonic measure is supported on a boundary subset of size approximately equal to $n^{\\beta_{\\alpha}}$, where the constant $\\beta_{\\alpha}\\in (0,\\frac{1}{\\alpha-1})$ depends only on the index $\\alpha$. Using an explicit expression of $\\beta_{\\alpha}$, we are able to show the uniform boundedness of $(\\beta_{\\alpha}, 1<\\alpha\\leq 2)$. These are generalizations of results in a recent paper of Curien and Le Gall (arXiv: 1304.7190).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-07T12:18:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60G50",
      "60K37"
    ],
    "keywords": [
      "infinite variance offspring distribution",
      "simple random walk",
      "large critical galton-watson trees",
      "harmonic measure refers",
      "uniform boundedness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.1583L"
    }
  }
}