{
  "id": "cond-mat/9804024",
  "version": "v1",
  "published": "1998-04-03T08:35:40.000Z",
  "updated": "1998-04-03T08:35:40.000Z",
  "title": "Level Spacing of Random Matrices in an External Source",
  "authors": [
    "E. Brezin",
    "S. Hikami"
  ],
  "comment": "25 pages, latex",
  "doi": "10.1103/PhysRevE.58.7176",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "In an earlier work we had considered a Gaussian ensemble of random matrices in the presence of a given external matrix source. The measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials, or on the Coulomb gas representation, are not available. Nevertheless the n-point correlation functions are still given in terms of the determinant of a kernel, known through an explicit integral representation. This kernel is no longer symmetric though and is not readily accessible to standard methods. In particular finding the level spacing probability is always a delicate problem in Fredholm theory, and we have to reconsider the problem within our model. We find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state. The problem is solved through coupled non-linear differential equations, which turn out to form a Hamiltonian system. As a result we find that the level spacing probability $p(s)$ behaves like $\\exp[ - C s^{{8\\over{3}}}]$ for large spacing $s$; this is consistent with the asymptotic behavior $\\exp[ - C s^{2 \\beta + 2}]$, whenever the density of state behaves near the edge as $\\rho(\\lambda)\\sim \\lambda^{\\beta}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-04-03T08:35:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random matrices",
      "external source",
      "level spacing probability",
      "longer unitary invariant",
      "coupled non-linear differential equations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Phys. Rev. E"
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}