{
  "id": "1406.6971",
  "version": "v2",
  "published": "2014-06-26T18:19:49.000Z",
  "updated": "2014-10-16T09:39:30.000Z",
  "title": "The minimum of a branching random walk outside the boundary case",
  "authors": [
    "Julien Barral",
    "Yueyun Hu",
    "Thomas Madaule"
  ],
  "comment": "35 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper is a complement to the studies on the minimum of a real-valued branching random walk. In the boundary case (Biggins, Kyprianou 2005), A\\\"{i}d\\'ekon in a seminal paper (2013) obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation when the log-generating function of the branching random walk explodes at some positive point and it cannot be reduced to the boundary case. In the associated thermodynamics framework this corresponds to a first order phase transition, while the boundary case corresponds to a second order phase transition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-26T18:19:49.000Z",
      "title": "The minimum of a branching random walk under a first order phase transition",
      "abstract": "This paper deals with a case left open in the studies of the asymptotic behavior of the minimum of a real-valued branching random walk. In the boundary case, Aid\\'ekon (2013) obtained the convergence in law of the minimum after a suitable renormalization. We study here the situation outside the Cram\\'er zone which cannot be reduced to the boundary case and in the associated thermodynamics framework corresponds to a first order phase transition, while the boundary case corresponds to a second order phase transition.",
      "comment": "32 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-16T09:39:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05"
    ],
    "keywords": [
      "first order phase transition",
      "branching random walk",
      "boundary case",
      "second order phase transition",
      "case left open"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.6971B"
    }
  }
}