{
  "id": "1012.3720",
  "version": "v1",
  "published": "2010-12-16T19:08:52.000Z",
  "updated": "2010-12-16T19:08:52.000Z",
  "title": "Enumeration of closed random walks in the square lattice according to their areas",
  "authors": [
    "Morteza Mohammad-Noori"
  ],
  "comment": "12 pages, 1 Figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study the area distribution of closed walks of length $n$, beginning and ending at the origin. The concept of area of a walk in the square lattice is generalized and the usefulness of the new concept is demonstrated through a simple argument. It is concluded that the number of walks of length $n$ and area $s$ equals to the coefficient of $z^s$ in the expression $(x+x^{-1}+y+y^{-1})^n$, where the calculations are performed in a special group ring $R[x,y,z]$. A polynomial time algorithm for calculating these values, is then concluded. Finally, the provided algorithm and the results of implementation are compared with previous works.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-16T19:08:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15"
    ],
    "keywords": [
      "closed random walks",
      "square lattice",
      "enumeration",
      "polynomial time algorithm",
      "simple argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3720M"
    }
  }
}