{
  "id": "1708.04223",
  "version": "v1",
  "published": "2017-08-14T17:39:53.000Z",
  "updated": "2017-08-14T17:39:53.000Z",
  "title": "Random walks on rings and modules",
  "authors": [
    "Arvind Ayyer",
    "Benjamin Steinberg"
  ],
  "categories": [
    "math.CO",
    "math.GR",
    "math.PR",
    "math.RA",
    "math.RT"
  ],
  "abstract": "We consider two natural models of random walks on a module $V$ over a finite commutative ring $R$ driven simultaneously by addition of random elements in $V$, and multiplication by random elements in $R$. In the coin-toss walk, either one of the two operations is performed depending on the flip of a coin. In the affine walk, random elements $a \\in R,b \\in V$ are sampled independently, and the current state $x$ is taken to $ax+b$. For both models, we obtain the complete spectrum of the transition matrix from the representation theory of the monoid of all affine maps on $V$ under a suitable hypothesis on the measure on $V$ (the measure on $R$ can be arbitrary).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-14T17:39:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "20M30",
      "13M99",
      "05E10",
      "60C05"
    ],
    "keywords": [
      "random walks",
      "random elements",
      "representation theory",
      "transition matrix",
      "natural models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}