{
  "id": "1412.7721",
  "version": "v1",
  "published": "2014-12-24T17:46:10.000Z",
  "updated": "2014-12-24T17:46:10.000Z",
  "title": "Asymptotic expansion of a partition function related to the sinh-model",
  "authors": [
    "G. Borot",
    "A. Guionnet",
    "K. K. Kozlowski"
  ],
  "comment": "132 pages, 4 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR",
    "nlin.SI"
  ],
  "abstract": "This paper develops a method to carry out the large-$N$ asymptotic analysis of a class of $N$-dimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size $N$, but in the present problem, two scales $1/N^{\\alpha}$ and $1/N$ naturally occur. In our case, the equilibrium measure is $N^{\\alpha}$-dependent and characterised by means of the solution to a $2\\times 2$ Riemann--Hilbert problem, whose large-$N$ behavior is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-$N$ behavior of the free energy explicitly up to $o(1)$. The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales $1/N^{\\alpha}$ and $1/N$, thus waiving the analyticity assumptions often used in random matrix theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-24T17:46:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partition function",
      "asymptotic expansion",
      "sinh-model",
      "asymptotic analysis",
      "random matrix theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 132,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1335883,
      "adsabs": "2014arXiv1412.7721B"
    }
  }
}