Search

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]

We present here an overview of the history, applications and important properties of a function which we refer to as the Lévy integral. For certain values of its characteristic parameter, the Lévy integral defines the symmetric Lévy stable probability density function. As we discuss however, the Lévy integral has applications to a number of other fields besides probability, including random matrix theory, number theory and asymptotics beyond all orders. We exhibit a direct relationship between the Lévy integral and a number theoretic series which we refer to as the generalized Euler-Jacobi series. The complete asymptotic expansions for all natural values of its parameter are presented, and in particular it is pointed out that the intricate exponentially small series become dominant for certain parameter values. [ABSTRACT FROM AUTHOR]

The complete asymptotic expansions, that is to say expansions which include any exponentially small terms lying beyond all orders of the asymptotic power series, are calculated for the Fermi-Dirac integrals. We present two methods to accomplish this, the first in the complex plane utilizing Mellin transforms and Hankel's representation of the gamma function, and the second on the real line using the known asymptotic expansions of the confluent hypergeometric functions. The complete expansions of F[sub p](η) are then used to investigate the effect that these traditionally neglected exponentially small terms have on physical systems. It is shown that for a 2 dimensional nonrelativistic ideal Fermi gas, the subdominant exponentially small series becomes dominant. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]