{
  "id": "math-ph/0510033",
  "version": "v3",
  "published": "2005-10-08T20:06:15.000Z",
  "updated": "2006-04-28T16:35:12.000Z",
  "title": "Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase",
  "authors": [
    "Pavel Bleher",
    "Vladimir Fokin"
  ],
  "comment": "53 pages, 8 figures",
  "doi": "10.1007/s00220-006-0097-y",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.CO",
    "math.MP"
  ],
  "abstract": "The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $N\\times N$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $N$ asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann-Hilbert approach and the Deift-Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn-Justin that the partition function of the six-vertex model with DWBC has the asymptotics, $Z_N\\sim CN^\\kappa e^{N^2f}$ as $N\\to\\infty$, and we find the exact value of the exponent $\\kappa$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2006-04-28T16:35:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B23"
    ],
    "keywords": [
      "domain wall boundary conditions",
      "six-vertex model",
      "exact solution",
      "disordered phase",
      "deift-zhou nonlinear steepest descent method"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Commun. Math. Phys."
    },
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}