{
  "id": "1410.6927",
  "version": "v1",
  "published": "2014-10-25T15:04:35.000Z",
  "updated": "2014-10-25T15:04:35.000Z",
  "title": "Scaling functions in the square Ising model",
  "authors": [
    "S. Hassani",
    "J-M. Maillard"
  ],
  "comment": "26 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We show and give the linear differential operators ${\\cal L}^{scal}_q$ of order q= n^2/4+n+7/8+(-1)^n/8, for the integrals $I_n(r)$ which appear in the two-point correlation scaling function of Ising model $ F_{\\pm}(r)= \\lim_{scaling} {\\cal M}_{\\pm}^{-2} < \\sigma_{0,0} \\, \\sigma_{M,N}> = \\sum_{n} I_{n}(r)$. The integrals $ I_{n}(r)$ are given in expansion around r= 0 in the basis of the formal solutions of $\\, {\\cal L}^{scal}_q$ with transcendental combination coefficients. We find that the expression $ r^{1/4}\\,\\exp(r^2/8)$ is a solution of the Painlev\\'e VI equation in the scaling limit. Combinations of the (analytic at $ r= 0$) solutions of $ {\\cal L}^{scal}_q$ sum to $ \\exp(r^2/8)$. We show that the expression $ r^{1/4} \\exp(r^2/8)$ is the scaling limit of the correlation function $ C(N, N)$ and $ C(N, N+1)$. The differential Galois groups of the factors occurring in the operators $ {\\cal L}^{scal}_q$ are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-25T15:04:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34M55",
      "47E05",
      "81Qxx",
      "32G34",
      "34Lxx",
      "34Mxx",
      "14Kxx"
    ],
    "keywords": [
      "square ising model",
      "transcendental combination coefficients",
      "two-point correlation scaling function",
      "linear differential operators",
      "painleve vi equation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/48/11/115205",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2015,
      "month": "Mar",
      "volume": 48,
      "number": 11,
      "pages": 115205
    },
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JPhA...48k5205H"
    }
  }
}