{
  "id": "1611.05728",
  "version": "v1",
  "published": "2016-11-17T15:11:00.000Z",
  "updated": "2016-11-17T15:11:00.000Z",
  "title": "Component structure of the configuration model: barely supercritical case",
  "authors": [
    "Remco van der Hofstad",
    "Svante Janson",
    "Malwina Luczak"
  ],
  "comment": "46 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $\\nu_n={\\mathbb E} [D_n(D_n-1)]/{\\mathbb E}[D_n]$ and, assuming that $\\nu_n \\to 1$ as $n \\to \\infty$, we write $\\varepsilon_n=\\nu_n-1$. We call the setting where $\\varepsilon_n n^{1/3}/({\\mathbb E}[D_n^3])^{2/3} \\to \\infty$ the {\\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \\to \\infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \\rho_n {\\mathbb E} D_n (1+o(1))$, where $\\rho_n$ denotes the survival probability of a branching process with offspring distribution $D_n^*-1$. This extends earlier results of Janson and Luczak~\\cite{JanLuc07}, as well as those of Janson, Luczak, Windridge and House~\\cite{SJ300} to the case where the third moment of $D_n$ is unbounded, filling the gap in the literature. We further study the size of the largest component in the \\emph{critical} regime, where $\\varepsilon_n = O(n^{-1/3} ({\\mathbb E} D_n^3)^{2/3})$, extending and complementing results of Hatami and Molloy~\\cite{HatamiMolloy}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-17T15:11:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60C05"
    ],
    "keywords": [
      "configuration model",
      "barely supercritical case",
      "component structure",
      "extends earlier results",
      "unique giant component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}