{
  "id": "1412.3781",
  "version": "v1",
  "published": "2014-12-11T19:59:25.000Z",
  "updated": "2014-12-11T19:59:25.000Z",
  "title": "Four Random Permutations Conjugated by an Adversary Generate $S_n$ with High Probability",
  "authors": [
    "Robin Pemantle",
    "Yuval Peres",
    "Igor Rivin"
  ],
  "comment": "19pages, 1 figure",
  "categories": [
    "math.PR",
    "cs.SC",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove a conjecture dating back to a 1978 paper of D.R.\\ Musser~\\cite{musserirred}, namely that four random permutations in the symmetric group $\\mathcal{S}_n$ generate a transitive subgroup with probability $p_n > \\epsilon$ for some $\\epsilon > 0$ independent of $n$, even when an adversary is allowed to conjugate each of the four by a possibly different element of $\\S_n$ (in other words, the cycle types already guarantee generation of $\\mathcal{S}_n$). This is closely related to the following random set model. A random set $M \\subseteq \\mathbb{Z}^+$ is generated by including each $n \\geq 1$ independently with probability $1/n$. The sumset $\\text{sumset}(M)$ is formed. Then at most four independent copies of $\\text{sumset}(M)$ are needed before their mutual intersection is no longer infinite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-11T19:59:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "12Y05",
      "68W20",
      "68W30",
      "68W40"
    ],
    "keywords": [
      "random permutations",
      "high probability",
      "adversary generate",
      "random set model",
      "cycle types"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.3781P"
    }
  }
}