{
  "id": "1905.00316",
  "version": "v1",
  "published": "2019-05-01T13:54:45.000Z",
  "updated": "2019-05-01T13:54:45.000Z",
  "title": "Large, lengthy graphs look locally like lines",
  "authors": [
    "Itai Benjamini",
    "Tom Hutchcroft"
  ],
  "comment": "8 pages",
  "categories": [
    "math.MG",
    "math.CO",
    "math.PR"
  ],
  "abstract": "We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a graph, there exists a mesoscopic scale on which the graph looks like $\\mathbb{R}$ in the sense that the rescaled ball is close to a line segment in the Gromov-Hausdorff metric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-01T13:54:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lengthy graphs look",
      "unimodular random rooted graphs",
      "line segment",
      "graph looks",
      "mesoscopic scale"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}