{
  "id": "1303.1130",
  "version": "v1",
  "published": "2013-03-05T18:32:27.000Z",
  "updated": "2013-03-05T18:32:27.000Z",
  "title": "Universality and critical behavior in the chiral two-matrix model",
  "authors": [
    "Steven Delvaux",
    "Dries Geudens",
    "Lun Zhang"
  ],
  "comment": "70 pages, 10 figures",
  "categories": [
    "math-ph",
    "hep-th",
    "math.CA",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the chiral two-matrix model with polynomial potential functions $V$ and $W$, which was introduced by Akemann, Damgaard, Osborn and Splittorff. We show that the squared singular values of each of the individual matrices in this model form a determinantal point process with correlation kernel determined by a matrix-valued Riemann-Hilbert problem. The size of the Riemann-Hilbert matrix depends on the degree of the potential function $W$ (or $V$ respectively). In this way we obtain the chiral analogue of a result of Kuijlaars-McLaughlin for the non-chiral two-matrix model. The Gaussian case corresponds to $V,W$ being linear. For the case where $W(y)=y^2/2+\\alpha y$ is quadratic, we derive the large $n$-asymptotics of the Riemann-Hilbert problem by means of the Deift-Zhou steepest descent method. This proves universality in this case. An important ingredient in the analysis is a third-order differential equation. Finally we show that if also $V(x)=x$ is linear, then a multi-critical limit of the kernel exists which is described by a $4\\times 4$ matrix-valued Riemann-Hilbert problem associated to the Painlev\\'e II equation $q\"(x) = xq(x)+2q^3(x)-\\nu-1/2$. In this way we obtain the chiral analogue of a recent result by Duits and the second author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-05T18:32:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical behavior",
      "universality",
      "matrix-valued riemann-hilbert problem",
      "deift-zhou steepest descent method",
      "chiral analogue"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/26/8/2231",
      "journal": "Nonlinearity",
      "year": 2013,
      "month": "Aug",
      "volume": 26,
      "number": 8,
      "pages": 2231
    },
    "note": {
      "typesetting": "TeX",
      "pages": 70,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1222540,
      "adsabs": "2013Nonli..26.2231D"
    }
  }
}