{
  "id": "1509.00256",
  "version": "v1",
  "published": "2015-09-01T12:26:28.000Z",
  "updated": "2015-09-01T12:26:28.000Z",
  "title": "On the joint behaviour of speed and entropy of random walks on groups",
  "authors": [
    "Gideon Amir"
  ],
  "comment": "12 pages",
  "categories": [
    "math.GR",
    "math.PR"
  ],
  "abstract": "For every $3/4\\le \\delta, \\beta< 1$ satisfying $\\delta\\leq \\beta < \\frac{1+\\delta}{2}$ we construct a finitely generated group $\\Gamma$ and a (symmetric, finitely supported) random walk $X_n$ on $\\Gamma$ so that its expected distance from its starting point satisfies $E|X_n|\\asymp n^{\\beta}$ and its entropy satisfies $H(X_n)\\asymp n^\\delta$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\\frac{3}{4}}\\leq h(n)\\leq f(n)\\leq \\sqrt{{nh(n)}/{\\log (n+1)}}\\leq n^\\gamma$ for some $\\gamma<1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-01T12:26:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C81",
      "20E08",
      "20F65",
      "60B15"
    ],
    "keywords": [
      "random walk",
      "joint behaviour",
      "entropy satisfies",
      "constant factor",
      "functions satisfy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}