{
  "id": "0806.1236",
  "version": "v1",
  "published": "2008-06-06T21:09:52.000Z",
  "updated": "2008-06-06T21:09:52.000Z",
  "title": "Monotone loop models and rational resonance",
  "authors": [
    "Alan Hammond",
    "Richard Kenyon"
  ],
  "comment": "22 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $T_{n,m}=\\mathbb Z_n\\times\\mathbb Z_m$, and define a random mapping $\\phi\\colon T_{n,m}\\to T_{n,m}$ by $\\phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random walks'' $\\phi$ in the limit $m,n\\to\\infty$, and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a rational $p/q$, we show that there are likely to be on the order of $\\sqrt{n}$ cycles, each of length O(n), whereas for $m/n$ far from any rational with small denominator, there are a bounded number of cycles, and for typical $m/n$ each cycle has length on the order of $n^{4/3}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-06-06T21:09:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60Gxx"
    ],
    "keywords": [
      "monotone loop models",
      "rational resonance",
      "equal probability",
      "orbit structure",
      "quenched random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.1236H"
    }
  }
}