{
  "id": "1302.2971",
  "version": "v2",
  "published": "2013-02-13T02:39:38.000Z",
  "updated": "2020-05-09T22:33:42.000Z",
  "title": "A Local Limit Theorem and Loss of Rotational Symmetry of Planar Symmetric Simple Random Walk",
  "authors": [
    "Christian Beneš"
  ],
  "comment": "20 pages, 3 figures",
  "journal": "Trans. Amer. Math. Soc. 371 (2019), 2553-2573",
  "doi": "10.1090/tran/7399",
  "categories": [
    "math.PR"
  ],
  "abstract": "We derive a local limit theorem for normal, moderate, and large deviations for symmetric simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the walk's starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$, where $x$ is thought of as dependent on $n$, in dimensions one and two for $P(S_n=x)$, the probability that symmetric simple random walk $S$ started at the origin is at some point $x$ at time $n$, that are valid for all $x$. We also show that the behavior of planar symmetric simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where the random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S_n|)$, $S_n$ is more likely to be found along the coordinate axes. In this paper, we give a description of how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-02-13T02:39:38.000Z",
      "title": "A Local Central Limit Theorem and Loss of Rotational Symmetry of Planar Simple Random Walk",
      "abstract": "We use elementary methods to derive a sharp local central limit theorem for simple random walk on the square lattice in dimensions one and two that is an improvement of existing results for points that are particularly distant from the starting point. More specifically, we give explicit asymptotic expressions in terms of $n$ and $x$ in dimensions one and two for $P(S(n)=x)$, the probability that simple random walk $S$ is at some point $x$ at time $n$, that are valid for all $|x|\\leq n/\\log^2(n)$. We also show that the behavior of planar simple random walk differs radically from that of planar standard Brownian motion outside of the disk of radius $n^{3/4}$, where simple random walk ceases to be approximately rotationally symmetric. Indeed, if $n^{3/4}=o(|S(n)|)$, S(n) is more likely to be found along the coordinate axes. This loss of rotational symmetry is not surprising, since if $|S(n)|=n$, there are only four possible locations for S(n). In this paper, we show how the transition from approximate rotational symmetry to complete concentration of $S$ along the coordinate axes occurs.",
      "comment": "12 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2020-05-09T22:33:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60F05"
    ],
    "keywords": [
      "local central limit theorem",
      "planar simple random walk",
      "rotational symmetry",
      "standard brownian motion outside",
      "simple random walk differs"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "AMS"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.2971B"
    }
  }
}