{
  "id": "cond-mat/0401209",
  "version": "v1",
  "published": "2004-01-13T10:42:33.000Z",
  "updated": "2004-01-13T10:42:33.000Z",
  "title": "Random Walks on Hyperspheres of Arbitrary Dimensions",
  "authors": [
    "Jean-Michel Caillol"
  ],
  "comment": "10 pages",
  "journal": "J. Phys. A: Math. Gen. 37, 3077 (2004).",
  "doi": "10.1088/0305-4470/37/9/001",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider random walks on the surface of the sphere $S_{n-1}$ ($n \\geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2 > \\varpropto t $ valid in $E_{n-1}$ should be replaced in $S_{n-1}$ by the generic law $<\\cos \\theta > \\varpropto \\exp(-t/\\tau)$, where $\\theta$ denotes the angular displacement of the walker. More generally one has $<C^{n/2-1}_{L}\\cos(\\theta)> \\varpropto \\exp(-t/ \\tau(L,n))$ where $C^{n/2-1}_{L}$ a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in $S_{n-1}$ are given tentatively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-13T10:42:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary dimensions",
      "hypersphere",
      "dimensional euclidean space",
      "conjectures concerning random walks",
      "usual law"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2004,
      "month": "Mar",
      "volume": 37,
      "number": 9,
      "pages": 3077
    },
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004JPhA...37.3077C"
    }
  }
}