{
  "id": "1502.04038",
  "version": "v1",
  "published": "2015-02-13T16:02:44.000Z",
  "updated": "2015-02-13T16:02:44.000Z",
  "title": "Random Walks on countable groups",
  "authors": [
    "Michael Björklund"
  ],
  "comment": "10 pages, no figures. Substantial overlap with the (longer) paper \"Five remarks about random walks on groups\", http://arxiv.org/abs/1406.0763",
  "categories": [
    "math.DS",
    "math.GR",
    "math.PR"
  ],
  "abstract": "We begin by giving a short and essentially self-contained proof of the equivalence between the vanishing of the drift of a finitely generated symmetric measured group with finite first moment and the absence of bounded harmonic functions; a result due to Kaimanovich-Vershik and Karlsson-Ledrappier. Given a measured group $(G,\\mu)$, we introduce the new notion of weak $(G,\\mu)$-mixing and show that the Poisson boundary is weakly $(G,\\mu)$-mixing. In particular, this gives a new proof of the fact that the \"double\" Poisson boundary is weakly mixing in the non-singular sense, which was first observed by Kaimanovich. Finally, we show that (non-singular) weak mixing for ergodic $(G,\\mu)$-spaces is equivalent to the absence of a probability measure preserving factor with discrete spectrum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-13T16:02:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walks",
      "countable groups",
      "generated symmetric measured group",
      "poisson boundary",
      "finite first moment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150204038B"
    }
  }
}