{
  "id": "1101.1816",
  "version": "v1",
  "published": "2011-01-10T13:54:38.000Z",
  "updated": "2011-01-10T13:54:38.000Z",
  "title": "The uniform measure on a Galton-Watson tree without the XlogX condition",
  "authors": [
    "elie aidekon"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a Galton--Watson tree with offspring distribution $\\nu$ of finite mean. The uniform measure on the boundary of the tree is obtained by putting mass $1$ on each vertex of the $n$-th generation and taking the limit $n\\to \\infty$. In the case $E[\\nu\\ln(\\nu)]<\\infty$, this measure has been well studied, and it is known that the Hausdorff dimension of the measure is equal to $\\ln(m)$ (\\cite{hawkes}, \\cite{lpp95}). When $E[\\nu \\ln(\\nu)]=\\infty$, we show that the dimension drops to $0$. This answers a question of Lyons, Pemantle and Peres \\cite{LyPemPer97}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-10T13:54:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "28A78"
    ],
    "keywords": [
      "galton-watson tree",
      "uniform measure",
      "xlogx condition",
      "finite mean",
      "th generation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}