{
  "id": "1310.1792",
  "version": "v2",
  "published": "2013-10-07T14:18:01.000Z",
  "updated": "2014-02-27T08:44:24.000Z",
  "title": "On hitting times for simple random walk on dense Erdös-Rényi random graphs",
  "authors": [
    "Matthias Löwe",
    "Felipe Torres"
  ],
  "comment": "9 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $G(N,p)=(V,E)$ be an Erd\\\"os-R\\'enyi random graph and $(X_n)_{n \\in \\mathbb{N}}$ be a simple random walk on it. We study the the order of magnitude of $\\sum_{i \\in V} \\pi_ih_{ij} $ where $\\pi_i=d_i / 2|E|$ for $d_i$ the number of neighbors of node $i$ and $h_{ij}$ the hitting time for $(X_n)_{n \\in \\mathbb{N}}$ between nodes $i$ and $j$, in a regime of $p=p(N)$ such that $G(N,p)$ is almost surely connected as $N\\to\\infty$. Our main result is that $\\sum_{i \\in V} \\pi_ih_{ij} $ is almost surely of order $N(1+o(1))$ as $N\\to \\infty$, which coincides with previous results in the physics literature \\cite{sood}, though our techniques are based on large deviations bounds on the number of neighbors of a typical node and the number of edges in $G(N,p)$ together with recent work on bounds on the spectrum of the (random) adjacency matrix of $G(N,p)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-02-27T08:44:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "05C81",
      "05C80"
    ],
    "keywords": [
      "dense erdös-rényi random graphs",
      "simple random walk",
      "hitting time",
      "large deviations bounds",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.1792L"
    }
  }
}