{
  "id": "2204.10096",
  "version": "v1",
  "published": "2022-04-21T13:42:13.000Z",
  "updated": "2022-04-21T13:42:13.000Z",
  "title": "Factorization of Ising correlations C(M,N) for $ ν= \\, -k$ and M+N odd, $M \\le N$, $T < T_c$ and their lambda extensions",
  "authors": [
    "S. Boukraa",
    "C. Cosgrove",
    "J. -M. Maillard",
    "B. M. McCoy"
  ],
  "comment": "45 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We study the factorizations of Ising low-temperature correlations C(M,N) for $\\nu=-k$ and M+N odd, $M \\le N$, for both the cases $M\\neq 0$ where there are two factors, and $M=0$ where there are four factors. We find that the two factors for $ M \\neq 0$ satisfy the same non-linear differential equation and, similarly, for M=0 the four factors each satisfy Okamoto sigma-form of Painlev\\'e VI equations with the same Okamoto parameters. Using a Landen transformation we show, for $M\\neq 0$, that the previous non-linear differential equation can actually be reduced to an Okamoto sigma-form of Painlev\\'e VI equation. For both the two and four factor case, we find that there is a one parameter family of boundary conditions on the Okamoto sigma-form of Painlev\\'e VI equations which generalizes the factorization of the correlations C(M,N) to an additive decomposition of the corresponding sigma's solutions of the Okamoto sigma-form of Painlev\\'e VI equation which we call lambda extensions. At a special value of the parameter, the lambda-extensions of the factors of C(M,N) reduce to homogeneous polynomials in the complete elliptic functions of the first and second kind. We also generalize some Tracy-Widom (Painlev\\'e V) relations between the sum and difference of sigma's to this Painlev\\'e VI framework.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-21T13:42:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34M55",
      "47E05",
      "81Qxx",
      "32G34",
      "34Lxx",
      "34Mxx",
      "14Kxx"
    ],
    "keywords": [
      "painleve vi equation",
      "lambda extensions",
      "ising correlations",
      "non-linear differential equation",
      "factorization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}