{
  "id": "0808.1256",
  "version": "v1",
  "published": "2008-08-08T17:58:42.000Z",
  "updated": "2008-08-08T17:58:42.000Z",
  "title": "Bulk asymptotics of skew-orthogonal polynomials for quartic double well potential and universality in the matrix model",
  "authors": [
    "Saugata Ghosh"
  ],
  "comment": "6 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We derive bulk asymptotics of skew-orthogonal polynomials (sop) $\\pi^{\\bt}_{m}$, $\\beta=1$, 4, defined w.r.t. the weight $\\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \\to\\infty$ there exists an $\\epsilon > 0$, such that $\\epsilon\\leq (m/N)\\leq \\lambda_{\\rm cr}-\\epsilon$, where $\\lambda_{\\rm cr}$ is the critical value which separates sop with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between sop and orthogonal polynomials (op) and using asymptotic results of op derived in \\cite{bleher}. Finally, we apply these asymptotic results of sop and their recursion coefficients in the generalized Christoffel-Darboux formula (GCD) \\cite{ghosh3} to obtain level densities and sine-kernels in the bulk of the spectrum for orthogonal and symplectic ensembles of random matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-08T17:58:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.-a",
      "02.50.-r",
      "02.10.Yn",
      "02.10.De"
    ],
    "keywords": [
      "skew-orthogonal polynomials",
      "bulk asymptotics",
      "matrix model",
      "quartic double",
      "universality"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.3093266",
      "journal": "Journal of Mathematical Physics",
      "year": 2009,
      "month": "Jun",
      "volume": 50,
      "number": 6,
      "pages": 3515
    },
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009JMP....50f3515G"
    }
  }
}