{
  "id": "1110.5203",
  "version": "v3",
  "published": "2011-10-24T11:44:45.000Z",
  "updated": "2012-05-28T06:37:59.000Z",
  "title": "Asymptotics of trees with a prescribed degree sequence and applications",
  "authors": [
    "Nicolas Broutin",
    "Jean-François Marckert"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $t$ be a rooted tree and $n_i(t)$ the number of nodes in $t$ having $i$ children. The degree sequence $(n_i(t),i\\geq 0)$ of $t$ satisfies $\\sum_{i\\ge 0} n_i(t)=1+\\sum_{i\\ge 0} in_i(t)=|t|$, where $|t|$ denotes the number of nodes in $t$. In this paper, we consider trees sampled uniformly among all trees having the same degree sequence $\\ds$; we write $`P_\\ds$ for the corresponding distribution. Let $\\ds(\\kappa)=(n_i(\\kappa),i\\geq 0)$ be a list of degree sequences indexed by $\\kappa$ corresponding to trees with size $\\nk\\to+\\infty$. We show that under some simple and natural hypotheses on $(\\ds(\\kappa),\\kappa>0)$ the trees sampled under $`P_{\\ds(\\kappa)}$ converge to the Brownian continuum random tree after normalisation by $\\nk^{1/2}$. Some applications concerning Galton--Watson trees and coalescence processes are provided.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-05-28T06:37:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "60C05",
      "60F17"
    ],
    "keywords": [
      "prescribed degree sequence",
      "asymptotics",
      "brownian continuum random tree",
      "applications concerning galton-watson trees",
      "coalescence processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.5203B"
    }
  }
}