{
  "id": "1811.00626",
  "version": "v1",
  "published": "2018-11-01T20:32:18.000Z",
  "updated": "2018-11-01T20:32:18.000Z",
  "title": "Action convergence of operators and graphs",
  "authors": [
    "Agnes Backhausz",
    "Balazs Szegedy"
  ],
  "categories": [
    "math.CO",
    "math.FA",
    "math.PR"
  ],
  "abstract": "We present a new approach to graph limit theory which unifies and generalizes the two most well developed directions, namely dense graph limits (even the more general $L^p$ limits) and Benjamini--Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called $P$-operators) of the form $L^\\infty(\\Omega)\\to L^1(\\Omega)$ for probability spaces $\\Omega$. We introduce a metric to compare $P$-operators (for example finite matrices) even if they act on different spaces. We prove a compactness result which implies that in appropriate norms, limits of uniformly bounded $P$-operators can again be represented by $P$-operators. We show that limits of operators representing graphs are self-adjoint, positivity-preserving $P$-operators called graphops. Graphons, $L^p$ graphons and graphings (known from graph limit theory) are special examples for graphops. We describe a new point of view on random matrix theory using our operator limit framework.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-01T20:32:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "action convergence",
      "graph limit theory",
      "operator limit framework",
      "random matrix theory",
      "dense graph limits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}