{
  "id": "1203.5624",
  "version": "v4",
  "published": "2012-03-26T10:41:34.000Z",
  "updated": "2014-08-26T17:01:04.000Z",
  "title": "On the scaling limit of finite vertex transitive graphs with large diameter",
  "authors": [
    "Itai Benjamini",
    "Hilary Finucane",
    "Romain Tessera"
  ],
  "comment": "Final version, to appear in Combinatorica",
  "categories": [
    "math.GR",
    "math.CO",
    "math.MG"
  ],
  "abstract": "Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\\bigl({diam(X_n)^{\\delta}}\\bigr)$ for $\\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-10-08T20:11:57.000Z",
      "abstract": "Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(\\diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $<q$, equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov's theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If $X_n$ is only roughly transitive and $|X_n| = o\\bigl({\\diam(X_n)^{\\delta}}\\bigr)$ for $\\delta > 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.",
      "comment": "We removed the assumption of bounded degree that appeared in the previous version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-08-26T17:01:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite vertex transitive graphs",
      "large diameter",
      "scaling limit",
      "gromov hausdorff distance",
      "invariant finsler metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1203.5624B"
    }
  }
}