{
  "id": "1402.5129",
  "version": "v2",
  "published": "2014-02-20T20:46:06.000Z",
  "updated": "2015-04-21T15:42:10.000Z",
  "title": "On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs",
  "authors": [
    "Julien Clancy",
    "Nathan Kaplan",
    "Timothy Leake",
    "Sam Payne",
    "Melanie Matchett Wood"
  ],
  "comment": "20 pages. v2: Improved exposition and appended code used to generate experimental evidence after the \\end{document} line in the source file. To appear in J. Algebraic Combin",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over the p-adic integers, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-20T20:46:06.000Z",
      "abstract": "We make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the group times the size of the group of automorphisms that preserve the pairing. We conjecture that the Jacobian of a random graph is cyclic with probability a little over .7935. We determine the values of several other statistics on Jacobians of random graphs that would follow from our conjectures. In support of the conjectures, we prove that random symmetric matrices over the p-adic integers, distributed according to Haar measure, have cokernels distributed according to the above heuristic. We also give experimental evidence in support of our conjectures.",
      "comment": "18 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-04-21T15:42:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "15B52"
    ],
    "keywords": [
      "random graph",
      "cohen-lenstra heuristic",
      "conjecture",
      "finite abelian group",
      "random symmetric matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.5129C"
    }
  }
}