{
  "id": "2401.06770",
  "version": "v1",
  "published": "2024-01-12T18:59:49.000Z",
  "updated": "2024-01-12T18:59:49.000Z",
  "title": "Random trees with local catastrophes: the Brownian case",
  "authors": [
    "Ariane Carrance",
    "Jérôme Casse",
    "Nicolas Curien"
  ],
  "comment": "31 pages, 7 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We introduce and study a model of plane random trees generalizing the famous Bienaym\\'e--Galton--Watson model but where births and deaths are locally correlated. More precisely, given a random variable $(B,H)$ with values in $\\{1,2,3, \\dots\\}^2$, given the state of the tree at some generation, the next generation is obtained (informally) by successively deleting $B$ individuals side-by-side and replacing them with $H$ new particles where the samplings are i.i.d. We prove that, in the critical case $\\mathbb{E}[B]=\\mathbb{E}[H]$, and under a third moment condition on $B$ and $H$, the random trees coding the genealogy of the population model converges towards the Brownian Continuum Random Tree. Interestingly, our proof does not use the classical height process or the {\\L}ukasiewicz exploration, but rather the stochastic flow point of view introduced by Bertoin and Le Gall.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-12T18:59:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B05",
      "60J80"
    ],
    "keywords": [
      "brownian case",
      "local catastrophes",
      "brownian continuum random tree",
      "stochastic flow point",
      "population model converges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}