{
  "id": "2211.02317",
  "version": "v1",
  "published": "2022-11-04T08:41:01.000Z",
  "updated": "2022-11-04T08:41:01.000Z",
  "title": "Conditioning (sub)critical L{é}vy trees by their maximal degree: Decomposition and local limit",
  "authors": [
    "Romain Abraham",
    "Jean-François Delmas",
    "Michel Nassif"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the maximal degree of (sub)critical L{\\'e}vy trees which arise as the scaling limits of Bienaym{\\'e}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{\\'e}vy tree conditioned to have a fixed maximal degree. In the case where the L{\\'e}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{\\'e}vy tree can be obtained by grafting a L{\\'e}vy forest on an independent size-biased L{\\'e}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{\\'e}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-04T08:41:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vy tree",
      "local limit",
      "large maximal degree converges",
      "uniformly chosen leaf",
      "poissonian decomposition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}