{
  "id": "0804.2039",
  "version": "v3",
  "published": "2008-04-13T02:46:04.000Z",
  "updated": "2008-08-11T12:37:36.000Z",
  "title": "Critical behavior and the limit distribution for long-range oriented percolation. II: Spatial correlation",
  "authors": [
    "Lung-Chi Chen",
    "Akira Sakai"
  ],
  "comment": "20 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that the Fourier transform of the properly-scaled normalized two-point function for sufficiently spread-out long-range oriented percolation with index \\alpha>0 converges to e^{-C|k|^{\\alpha\\wedge2}} for some C\\in(0,\\infty) above the upper-critical dimension 2(\\alpha\\wedge2). This answers the open question remained in the previous paper [arXiv:math/0703455]. Moreover, we show that the constant C exhibits crossover at \\alpha=2, which is a result of interactions among occupied paths. The proof is based on a new method of estimating fractional moments for the spatial variable of the lace-expansion coefficients.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-08-11T12:37:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B27"
    ],
    "keywords": [
      "limit distribution",
      "spatial correlation",
      "critical behavior",
      "sufficiently spread-out long-range oriented percolation",
      "properly-scaled normalized two-point function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.2039C"
    }
  }
}