{
  "id": "1411.4406",
  "version": "v1",
  "published": "2014-11-17T09:56:39.000Z",
  "updated": "2014-11-17T09:56:39.000Z",
  "title": "The two-point function of bicolored planar maps",
  "authors": [
    "Éric Fusy",
    "Emmanuel Guitter"
  ],
  "comment": "43 pages, 21 figures",
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We compute the distance-dependent two-point function of vertex-bicolored planar maps, i.e., maps whose vertices are colored in black and white so that no adjacent vertices have the same color. By distance-dependent two-point function, we mean the generating function of these maps with both a marked oriented edge and a marked vertex which are at a prescribed distance from each other. As customary, the maps are enumerated with arbitrary degree-dependent face weights, but the novelty here is that we also introduce color-dependent vertex weights. Explicit expressions are given for vertex-bicolored maps with bounded face degrees in the form of ratios of determinants of fixed size. Our approach is based on a slice decomposition of maps which relates the distance-dependent two-point function to the coefficients of the continued fraction expansions of some distance-independent map generating functions. Special attention is paid to the case of vertex-bicolored quadrangulations and hexangulations, whose two-point functions are also obtained in a more direct way involving equivalences with hard dimer statistics. A few consequences of our results, as well as some extension to vertex-tricolored maps, are also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-17T09:56:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distance-dependent two-point function",
      "arbitrary degree-dependent face weights",
      "color-dependent vertex weights",
      "hard dimer statistics",
      "distance-independent map generating functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.4406F"
    }
  }
}