{
  "id": "2106.12519",
  "version": "v1",
  "published": "2021-06-23T16:36:19.000Z",
  "updated": "2021-06-23T16:36:19.000Z",
  "title": "Poisson statistics and localization at the spectral edge of sparse Erdős--Rényi graphs",
  "authors": [
    "Johannes Alt",
    "Raphael Ducatez",
    "Antti Knowles"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the adjacency matrix $A$ of the Erd{\\H o}s--R\\'enyi graph on $N$ vertices with edge probability $d/N$. For $(\\log \\log N)^4 \\ll d \\lesssim \\log N$, we prove that the eigenvalues near the spectral edge form asymptotically a Poisson process and the associated eigenvectors are exponentially localized. As a corollary, at the critical scale $d \\asymp \\log N$, the limiting distribution of the largest nontrivial eigenvalue does not match with any previously known distribution. Together with [5], our result establishes the coexistence of a fully delocalized phase and a fully localized phase in the spectrum of $A$. The proof relies on a three-scale rigidity argument, which characterizes the fluctuations of the eigenvalues in terms of the fluctuations of sizes of spheres of radius 1 and 2 around vertices of large degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-23T16:36:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "15B52",
      "05C80"
    ],
    "keywords": [
      "sparse erdős-rényi graphs",
      "poisson statistics",
      "localization",
      "three-scale rigidity argument",
      "spectral edge form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}