{
  "id": "math/0612585",
  "version": "v2",
  "published": "2006-12-20T10:44:46.000Z",
  "updated": "2007-01-04T10:52:47.000Z",
  "title": "Volume growth and heat kernel estimates for the continuum random tree",
  "authors": [
    "David Croydon"
  ],
  "journal": "Probability Theory and Related Fields 140 (2008), no. 1-2, 207-238",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this article, we prove global and local (point-wise) volume and heat kernel bounds for the continuum random tree. We demonstrate that there are almost-surely logarithmic global fluctuations and log-logarithmic local fluctuations in the volume of balls of radius $r$ about the leading order polynomial term as $r\\to0$. We also show that the on-diagonal part of the heat kernel exhibits corresponding global and local fluctuations as $t\\to0$ almost-surely. Finally, we prove that this quenched (almost-sure) behaviour contrasts with the local annealed (averaged over all realisations of the tree) volume and heat kernel behaviour, which is smooth.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-01-04T10:52:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum random tree",
      "heat kernel estimates",
      "volume growth",
      "heat kernel behaviour",
      "almost-surely logarithmic global fluctuations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12585C"
    }
  }
}