Search

A bordering of Gaussian unitary ensemble matrices is considered, in which the bordered row consists of zero mean complex Gaussians N[0, σ/2] + iN[0, σ/2] off the diagonal and the real Gaussian N[formula] on the diagonal. We compute the explicit form of the eigenvalue probability function for such matrices as well as that for matrices obtained by repeating the bordering. The correlations are in general determinantal, and in the single bordering case the explicit form of the correlation kernel is computed. In the large N limit it is shown that μ and/or σ can be tuned to induce a separation of the largest eigenvalue. This effect is shown to be controlled by a single parameter, universal correlation kernel. [ABSTRACT FROM AUTHOR]

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s>1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s→∞, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogs, an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained. [ABSTRACT FROM AUTHOR]

In a recent study we have obtained correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, a matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically. [ABSTRACT FROM AUTHOR]

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N×N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk. [ABSTRACT FROM AUTHOR]

The density matrix for the impenetrable Bose gas in Dirichlet and Neumann boundary conditions can be written in terms of &land;Π[SUBl=1][SUPn]|cos ø[SUB1]-cos&thetas; [SUBl]|| cos ø [SUB2]-cos &thetas;[SUBl]|&rand, where the average is with respect to the eigenvalue probability density function for random unitary matrices from the classical groups Sp(n) and O[SUP+] (2n), respectively. In the large n limit log-gas considerations imply that the average factorizes into the product of averages of the form &land;Π[SUBl=1][SUPn]|cos ø- cos &thetas;[SUBl]|&rand. By changing variables this average in turn is a special case of the function of t obtained by averaging Π[SUBl=1][SUPn]|t-x[SUBl]|[SUP2q] over the Jacobi unitary ensemble from random matrix theory. The latter task is accomplished by a duality formula from the theory of Selberg correlation integrals, and the large n asymptotic form is obtained. The corresponding large n asymptotic form of the density matrix is used, via the exact solution of a particular integral equation, to compute the asymptotic form of the low lying effective single particle states and their occupations, which are proportional to √N. [ABSTRACT FROM AUTHOR]