{
  "id": "1802.00862",
  "version": "v1",
  "published": "2018-02-02T21:53:35.000Z",
  "updated": "2018-02-02T21:53:35.000Z",
  "title": "Projections of the Aldous chain on binary trees: Intertwining and consistency",
  "authors": [
    "Noah Forman",
    "Soumik Pal",
    "Douglas Rizzolo",
    "Matthias Winkel"
  ],
  "comment": "30 pages, 8 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the Aldous Markov chain on the space of rooted binary trees with $n$ labeled leaves in which at each transition a uniform random leaf is deleted and reattached to a uniform random edge. Now, fix $1\\le k < n$ and project the leaf mass onto the subtree spanned by the first $k$ leaves. This yields a binary tree with edge weights that we call a \"decorated $k$-tree with total mass $n$.\" We introduce label swapping dynamics for the Aldous chain so that, when it runs in stationarity, the decorated $k$-trees evolve as Markov chains themselves, and are projectively consistent over $k\\le n$. The construction of projectively consistent chains is a crucial step in the construction of the Aldous diffusion on continuum trees by the present authors, which is the $n\\rightarrow \\infty$ continuum analogue of the Aldous chain and will be taken up elsewhere. Some of our results have been generalized to Ford's alpha model trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-02T21:53:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60J80",
      "60J10"
    ],
    "keywords": [
      "binary tree",
      "aldous chain",
      "fords alpha model trees",
      "markov chain",
      "consistency"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}