{
  "id": "1908.04403",
  "version": "v1",
  "published": "2019-08-12T21:38:07.000Z",
  "updated": "2019-08-12T21:38:07.000Z",
  "title": "On breadth-first constructions of scaling limits of random graphs and random unicellular maps",
  "authors": [
    "Grégory Miermont",
    "Sanchayan Sen"
  ],
  "comment": "37 pages, 7 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We give alternate constructions of (i) the scaling limit of the uniform connected graphs with given fixed surplus, and (ii) the continuum random unicellular map (CRUM) of a given genus that start with a suitably tilted Brownian continuum random tree and make `horizontal' point identifications, at random heights, using the local time measures. Consequently, this can be seen as a continuum analogue of the breadth-first construction of a finite connected graph. In particular, this yields a breadth-first construction of the scaling limit of the critical Erd\\H{o}s-R\\'enyi random graph which answers a question posed in [2]. As a consequence of this breadth-first construction we obtain descriptions of the radii, the distance profiles, and the two point functions of these spaces in terms of functionals of tilted Brownian excursions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-12T21:38:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "breadth-first construction",
      "scaling limit",
      "random graph",
      "brownian continuum random tree",
      "tilted brownian continuum random"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}