{
  "id": "1409.7055",
  "version": "v1",
  "published": "2014-09-24T19:50:15.000Z",
  "updated": "2014-09-24T19:50:15.000Z",
  "title": "Liouville quantum gravity as a mating of trees",
  "authors": [
    "Bertrand Duplantier",
    "Jason Miller",
    "Scott Sheffield"
  ],
  "comment": "211 pages, approximately 50 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "There is a simple way to \"glue together\" a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the \"interface\" between the trees). We present an explicit and canonical way to embed the sphere in ${\\mathbf C} \\cup \\{ \\infty \\}$. In this embedding, the measure is Liouville quantum gravity (LQG) with parameter $\\gamma \\in (0,2)$, and the curve is space-filling SLE$_{\\kappa'}$ with $\\kappa' = 16/\\gamma^2$. Achieving this requires us to develop an extensive suite of tools for working with LQG surfaces. We explain how to conformally weld so-called \"quantum wedges\" to obtain new quantum wedges of different weights. We construct finite-volume quantum disks and spheres of various types, and give a Poissonian description of the set of quantum disks cut off by a boundary-intersecting SLE$_{\\kappa}(\\rho)$ process with $\\kappa \\in (0,4)$. We also establish a L\\'evy tree description of the set of quantum disks to the left (or right) of an SLE$_{\\kappa'}$ with $\\kappa' \\in (4,8)$. We show that given two such trees, sampled independently, there is a.s. a canonical way to \"zip them together\" and recover the SLE$_{\\kappa'}$. The law of the CRT pair we study was shown in an earlier paper to be the scaling limit of the discrete tree/dual-tree pair associated to an FK-decorated random planar map (RPM). Together, these results imply that FK-decorated RPM scales to CLE-decorated LQG in a certain \"tree structure\" topology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-24T19:50:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville quantum gravity",
      "construct finite-volume quantum disks",
      "quantum wedges",
      "quantum disks cut",
      "continuum random trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 211,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1318879,
      "adsabs": "2014arXiv1409.7055D"
    }
  }
}