{
  "id": "1107.2340",
  "version": "v3",
  "published": "2011-07-12T16:26:44.000Z",
  "updated": "2012-10-07T08:12:33.000Z",
  "title": "On the functions counting walks with small steps in the quarter plane",
  "authors": [
    "Irina Kurkova",
    "Kilian Raschel"
  ],
  "comment": "40 pages, 17 figures",
  "journal": "Publications math\\'ematiques de l'IH\\'ES: Volume 116, Issue 1 (2012), Pages 69-114",
  "doi": "10.1007/s10240-012-0045-7",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Models of spatially homogeneous walks in the quarter plane ${\\bf Z}_+^{2}$ with steps taken from a subset $\\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function $(x,y,z)\\mapsto Q(x,y;z)$ of the numbers $q(i,j;n)$ of such walks starting at the origin and ending at $(i,j) \\in {\\bf Z}_+^{2}$ after $n$ steps is studied. For all non-singular models of walks, the functions $x \\mapsto Q(x,0;z)$ and $y\\mapsto Q(0,y;z)$ are continued as multi-valued functions on ${\\bf C}$ having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of ${\\bf C}^2$, the interval $]0,1/|\\mathcal{S}|[$ of variation of $z$ splits into two dense subsets such that the functions $x \\mapsto Q(x,0;z)$ and $y\\mapsto Q(0,y;z)$ are shown to be holonomic for any $z$ from the one of them and non-holonomic for any $z$ from the other. This entails the non-holonomy of $(x,y,z)\\mapsto Q(x,y;z)$, and therefore proves a conjecture of Bousquet-M\\'elou and Mishna.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-10-07T08:12:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "30F10",
      "30D05"
    ],
    "keywords": [
      "functions counting walks",
      "quarter plane",
      "small steps",
      "dense subsets",
      "birational transformations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2340K"
    }
  }
}