{
  "id": "1609.06627",
  "version": "v1",
  "published": "2016-09-21T16:41:18.000Z",
  "updated": "2016-09-21T16:41:18.000Z",
  "title": "How round are the complementary components of planar Brownian motion?",
  "authors": [
    "Nina Holden",
    "Serban Nacu",
    "Yuval Peres",
    "Thomas S. Salisbury"
  ],
  "comment": "28 pages, 11 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a Brownian motion $W$ in ${\\bf C}$ started from $0$ and run for time 1. Let $A(1),A(2),\\dots$ denote the bounded connected components of ${\\bf C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\\in\\bf N$. Our main result is that ${\\bf E}[\\sum_i R(i)^2|\\log R(i)|^\\theta ]<\\infty$ for any $\\theta<1$. We also prove that $\\sum_i r(i)^2|\\log r(i)|=\\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-21T16:41:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05"
    ],
    "keywords": [
      "planar brownian motion",
      "complementary components",
      "round shape",
      "main result",
      "out-radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}