{
  "id": "1605.02868",
  "version": "v1",
  "published": "2016-05-10T06:54:04.000Z",
  "updated": "2016-05-10T06:54:04.000Z",
  "title": "Critical window for the configuration model: finite third moment degrees",
  "authors": [
    "Souvik Dhara",
    "Remco van der Hofstad",
    "Johan S. H. van Leeuwaarden",
    "Sanchayan Sen"
  ],
  "comment": "32 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the component sizes are $O(n^{2/3})$ and the re-scaled component sizes converge to the excursions of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in finite dimensions. This behavior was first observed for Erd\\H{o}s-R\\'enyi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-10T06:54:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite third moment degrees",
      "configuration model",
      "critical window",
      "finite third moment assumption",
      "asymptotic degree distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}