{
  "id": "math/0107068",
  "version": "v1",
  "published": "2001-07-10T09:34:48.000Z",
  "updated": "2001-07-10T09:34:48.000Z",
  "title": "Random Electrical Networks on Complete Graphs II: Proofs",
  "authors": [
    "Geoffrey Grimmett",
    "Harry Kesten"
  ],
  "comment": "51 pages, 9 figures. This paper was first circulated in 1983 and has never been submitted to a journal",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper contains the proofs of Theorems 2 and 3 of the article entitled Random Electrical Networks on Complete Graphs, written by the same authors and published in the Journal of the London Mathematical Society, vol. 30 (1984), pp. 171-192. The current paper was written in 1983 but was not published in a journal, although its existence was announced in the LMS paper. This TeX version was created on 9 July 2001. It incorporates minor improvements to formatting and punctuation, but no change has been made to the mathematics. We study the effective electrical resistance of the complete graph $K_{n+2}$ when each edge is allocated a random resistance. These resistances are assumed independent with distribution $P(R=\\infty)=1-n^{-1}\\gamma(n)$, $P(R\\le x) = n^{-1}\\gamma(n)F(x)$ for $0\\le x < \\infty$, where $F$ is a fixed distribution function and $\\gamma(n)\\to\\gamma\\ge 0$ as $n\\to\\infty$. The asymptotic effective resistance between two chosen vertices is identified in the two cases $\\gamma\\le 1$ and $\\gamma>1$, and the case $\\gamma=\\infty$ is considered. The analysis proceeds via detailed estimates based on the theory of branching processes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-07-10T09:34:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43"
    ],
    "keywords": [
      "complete graph",
      "resistance",
      "incorporates minor improvements",
      "article entitled random electrical networks",
      "analysis proceeds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......7068G"
    }
  }
}