{
  "id": "2101.08242",
  "version": "v1",
  "published": "2021-01-20T18:30:35.000Z",
  "updated": "2021-01-20T18:30:35.000Z",
  "title": "Sparse expanders have negative Ollivier-Ricci curvature",
  "authors": [
    "Justin Salez"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR",
    "cs.DM",
    "math.CO",
    "math.FA"
  ],
  "abstract": "We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To prove this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible \"at infinity\". We then transfer this result to finite graphs via local weak convergence and a relative compactness argument. We believe that this \"local weak limit\" approach to mixing properties of Markov chains will have many other applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-20T18:30:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "05C48",
      "37A30",
      "37A35"
    ],
    "keywords": [
      "sparse expanders",
      "local weak limit",
      "local weak convergence",
      "stationary random graphs",
      "non-negative ollivier-ricci curvature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}