{
  "id": "math/0509524",
  "version": "v1",
  "published": "2005-09-22T16:39:46.000Z",
  "updated": "2005-09-22T16:39:46.000Z",
  "title": "Continuum tree limit for the range of random walks on regular trees",
  "authors": [
    "Thomas Duquesne"
  ],
  "comment": "42 pages; 1 figure; 2004",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $b$ be an integer greater than 1 and let $W^{\\ee}=(W^{\\ee}_n; n\\geq 0)$ be a random walk on the $b$-ary rooted tree $\\U_b$, starting at the root, going up (resp. down) with probability $1/2+\\epsilon$ (resp. $1/2 -\\epsilon$), $\\epsilon \\in (0, 1/2)$, and choosing direction $i\\in \\{1, ..., b\\}$ when going up with probability $a_i$. Here $\\aa =(a_1, ..., a_b)$ stands for some non-degenerated fixed set of weights. We consider the range $\\{W^{\\ee}_n ; n\\geq 0 \\}$ that is a subtree of $\\U_b $. It corresponds to a unique random rooted ordered tree that we denote by $\\tau_{\\epsilon}$. We rescale the edges of $\\tau_{\\epsilon}$ by a factor $\\ee $ and we let $\\ee$ go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor $\\gamma (\\aa)$. More precisely, we prove that $\\tau_{\\epsilon}$ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by $\\gamma (\\aa)$. We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node ($b=\\infty$) and for a general set of weights $\\aa =(a_n, n\\geq 0)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-09-22T16:39:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "G22"
    ],
    "keywords": [
      "random walk",
      "continuum tree limit",
      "regular trees",
      "independent brownian motions",
      "deterministic phenomenon taken"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9524D"
    }
  }
}