{
  "id": "1809.07587",
  "version": "v1",
  "published": "2018-09-20T12:16:14.000Z",
  "updated": "2018-09-20T12:16:14.000Z",
  "title": "Emergence of extended states at zero in the spectrum of sparse random graphs",
  "authors": [
    "Simon Coste",
    "Justin Salez"
  ],
  "comment": "18 pages with 4 figures. Comments are welcome",
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "We confirm the long-standing prediction that $c=e\\approx 2.718$ is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erd\\H{o}s-Renyi random graph with average degree $c$. This is achieved by a detailed second-order analysis of the resolvent $(A-z)^{-1}$ near the singular point $z=0$, where $A$ is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring $c$. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-20T12:16:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sparse random graphs",
      "extended states",
      "arbitrary unimodular galton-watson trees",
      "random graph models",
      "adjacency operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}