{
  "id": "1408.2924",
  "version": "v3",
  "published": "2014-08-13T06:52:01.000Z",
  "updated": "2017-10-10T01:39:13.000Z",
  "title": "Reversing the cut tree of the Brownian continuum random tree",
  "authors": [
    "Nicolas Broutin",
    "Minmin Wang"
  ],
  "comment": "22 pages, 13 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the Aldous--Pitman fragmentation process [Ann Probab, 26(4):1703--1726, 1998] of a Brownian continuum random tree ${\\cal T}^{\\mathrm{br}}$. The associated cut tree cut$({\\cal T}^{\\mathrm{br}})$, introduced by Bertoin and Miermont [Ann Appl Probab, 23:1469--1493, 2013], is defined in a measurable way from the fragmentation process, describing the genealogy of the fragmentation, and is itself distributed as a Brownian CRT. In this work, we introduce a shuffle transform, which can be considered as the reverse of the map taking ${\\cal T}^{\\mathrm{br}}$ to cut$({\\cal T}^{\\mathrm{br}})$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-16T06:37:03.000Z",
      "abstract": "Consider the logging process of the Brownian continuum random tree (CRT) $\\cal T$ using a Poisson point process of cuts on its skeleton [Aldous and Pitman, Ann. Probab., vol. 26, pp. 1703--1726, 1998]. Then, the cut tree introduced by Bertoin and Miermont describes the genealogy of the fragmentation of $\\cal T$ into connected components [Ann. Appl. Probab., vol. 23, pp. 1469--1493, 2013]. This cut tree cut$(\\cal T)$ is distributed as another Brownian CRT, and is a function of the original tree $\\cal T$ and of the randomness in the logging process. We are interested in reversing the transformation of $\\cal T$ into cut$(\\cal T)$: we define a shuffling operation, which given a Brownian CRT $\\cal H$, yields another one shuff$(\\cal H)$ distributed in such a way that $(\\cal T$,cut$(\\cal T))$ and $($shuff$(\\cal H), \\cal H)$ have the same distribution.",
      "comment": "24 pages, 5 figures",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2017-10-10T01:39:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05"
    ],
    "keywords": [
      "brownian continuum random tree",
      "brownian crt",
      "poisson point process",
      "logging process",
      "cut tree cut"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.2924B"
    }
  }
}