{
  "id": "1610.05331",
  "version": "v1",
  "published": "2016-10-17T20:09:50.000Z",
  "updated": "2016-10-17T20:09:50.000Z",
  "title": "Self-similar real trees defined as fixed-points and their geometric properties",
  "authors": [
    "Nicolas Broutin",
    "Henning Sulzbach"
  ],
  "comment": "47 pages, 9 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence, the uniqueness of the fixed-points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed-points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially powerful to treat cases where the natural mass measure on the real tree only provides weak estimates on the Hausdorff dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-17T20:09:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-similar real trees",
      "geometric properties",
      "measured compact metric spaces",
      "fixed-point",
      "charging measured compact metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}