{
  "id": "2009.10544",
  "version": "v1",
  "published": "2020-09-22T13:34:36.000Z",
  "updated": "2020-09-22T13:34:36.000Z",
  "title": "Random walks, word metric and orbits distribution on the plane",
  "authors": [
    "Uriya Pumerantz"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "Given a countably infinite group $G$ acting on some space $X$, an increasing family of finite subsets $G_n$ and $x\\in X$, a natural question to ask is what asymptotical distribution the sets $G_nx$ form. More formally, we define for a function $f$ over $X$ the sums $S_n(f,x)=\\sum_{g\\in G_n}f(gx)$ and ask whether exists a function $\\Psi(n):\\mathbb{N}\\to\\mathbb{R}$ such that the sequence $\\Psi(n)S_n(f,x)$ converges. This is a delicate problem that was studied under various settings. We first show a full solution when elements are chosen using a carefully chosen word metric from a specific lattice in $SL(2,\\mathbb{Z})$ acting on the circle. In addition, it is proven that the resulting measure is stationary with respect to a certain random walk and has a tight connection to a well studied function from the field of Diophantine approximations. We then proceed to study the asymptotic distribution problem when elements are chosen using a random walk over $SL(2,\\mathbb{R})$ acting on $\\mathbb{R}^2$. We offer a variant of our initial problem which yields some surprising and interesting results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-22T13:34:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walk",
      "orbits distribution",
      "chosen word metric",
      "asymptotic distribution problem",
      "tight connection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}