{
  "id": "1608.00956",
  "version": "v1",
  "published": "2016-08-02T19:41:18.000Z",
  "updated": "2016-08-02T19:41:18.000Z",
  "title": "Convergence of the self-avoiding walk on random quadrangulations to SLE$_{8/3}$ on $\\sqrt{8/3}$-Liouville quantum gravity",
  "authors": [
    "Ewain Gwynne",
    "Jason Miller"
  ],
  "comment": "63 pages and 12 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.CV",
    "math.MP"
  ],
  "abstract": "Let $(Q_{\\mathrm{zip}} , \\lambda_{\\mathrm{zip}})$ be a uniform infinite quadrangulation of the half-plane decorated by a self-avoiding walk (SAW). We prove that $(Q_{\\mathrm{zip}} , \\lambda_{\\mathrm{zip}})$ converges in the scaling limit to the metric gluing of two independent Brownian half-planes identified along their positive boundary rays. Combined with other work of the authors, this implies the convergence of the SAW on a random quadrangulation to SLE$_{8/3}$ on a certain $\\sqrt{8/3}$-Liouville quantum gravity surface. The topology of convergence is the local Gromov-Hausdorff-Prokhorov-uniform topology, the natural generalization of the local Gromov-Hausdorff topology to curve-decorated metric measure spaces. We also prove analogous scaling limit results for uniform infinite quadrangulations of the whole plane decorated by either a one-sided or two-sided SAW.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-02T19:41:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random quadrangulation",
      "self-avoiding walk",
      "uniform infinite quadrangulation",
      "convergence",
      "liouville quantum gravity surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}