E. Kanzieper

TWO-BAND RANDOM MATRICES

Spectral correlations in unitary invariant, non-Gaussian ensembles of large random matrices possessing an eigenvalue gap are studied within the framework of the orthogonal polynomial technique. Both local and global characteristics of spectra are directly reconstructed from the recurrence equation for orthogonal polynomials associated with a given random matrix ensemble. It is established that an eigenvalue gap does not affect the local eigenvalue correlations which follow the universal sine and the universal multicritical laws in the bulk and soft-edge scaling limits, respectively. By contrast, global smoothed eigenvalue correlations do reflect the presence of a gap, and are shown to satisfy a new universal law exhibiting a sharp dependence on the odd/even dimension of random matrices whose spectra are bounded. In the case of unbounded spectrum, the corresponding universal `density-density correlator is conjectured to be generic for chaotic systems with a forbidden gap and broken time reversal symmetry.

Light scattering from slightly rough dielectric films

It is shown that the long-scale (smooth) component of the roughness spectrum of a slightly rough dielectric layer critically affects the angular distribution of radiation scattered from the surface. The interference pattern obtained from a sample with only small-scale roughness differs drastically from a sample with the same small-scale roughness but possessing slight (of the order λ/10) variation of the thickness of the dielectric layer. It is shown that when interference phenomena are significant and the dielectric film has long-scale roughness, conventional perturbation theory is invalid, even if the rms of roughness is much smaller than the wavelength. A model is presented that correctly predicts the measured angular intensity distributions in the scattered-light field for samples that possess arbitrary scales of roughness.

UNITARY RANDOM-MATRIX ENSEMBLE WITH GOVERNABLE LEVEL CONFINEMENT

A family of unitary ensuremath{alpha} ensembles of random matrices with governable confinement potential V(x)ensuremath{sim}ensuremath{Vert}x

Universality in invariant random-matrix models: Existence near the soft edge

{mathrm{ensuremath{Vert}}}^{mathrm{ensuremath{alpha}}}

Random-matrix models with the logarithmic-singular level confinement: Method of fictitious fermions

is studied employing exact results of the theory of nonclassical orthogonal polynomials. The density of levels, two-point kernel, locally rescaled two-level cluster function, and smoothed connected correlations between the density of eigenvalues are calculated for strong (ensuremath{alpha}g1) and border (ensuremath{alpha}=1) level confinement. It is shown that the density of states is a smooth function for ensuremath{alpha}g1, and has a well-pronounced peak at the band center for ensuremath{alpha}ensuremath{le}1. The case of border level confinement associated with transition point ensuremath{alpha}=1 is reduced to the exactly solvable Pollaczek random-matrix ensemble. Unlike the density of states, all the two-point correlators remain (after proper rescaling) universal down to and including ensuremath{alpha}=1. textcopyright{} 1996 The American Physical Society.

Theory of random matrices with strong level confinement: Orthogonal polynomial approach.

We consider two non-Gaussian ensembles of large Hermitian random matrices with strong level confinement and show that near the soft edge of the spectrum both scaled density of states and eigenvalue correlations follow so-called Airy laws inherent in Gaussian unitary ensemble. This suggests that the invariant one-matrix models should display universal eigenvalue correlations in the soft-edge scaling limit.

Interference in light scattering from slightly rough dielectric layers.

Abstract A joint distribution function of N eigenvalues of a U(N) invariant random-matrix ensemble can be interpreted as a probability density of finding N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture, a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic log-singular level confinement. An effective one-particle Schrodinger equation for wavefunctions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.

Reflection of waves from a strong scatterer buried in a random medium

Unitary ensembles of large N x N random matrices with a non-Gaussian probability distribution P[H] ~ exp{-TrV[H]} are studied using a theory of polynomials orthogonal with respect to exponential weights. Asymptotically exact expressions for density of levels, one- and two-point Greens functions are calculated. We show that in the large-N limit the properly rescaled local eigenvalue correlations are independent of P[H] while global smoothed connected correlations depend on P[H] only through the endpoints of spectrum. We also establish previously unknown intimate connection between structure of Szego function entering strong polynomial asymptotics and mean-field equation by Dyson.

Random matrix models with log-singular level confinement: method of fictitious fermions

The scattering of light from a slightly rough surface overlying a reflecting surface is investigated. It is shown that the long-scale component of the roughness spectrum plays a critical role in the scattering patterns obtained. The scattered interference patterns are critically dependent on small variation of the rms height of the long-scale component of the roughness. Conventional perturbation theory is found to be invalid in cases in which interference phenomena in the scattering are of importance. A model is proposed that quantitatively describes the measured angular intensity distributions.

Interference in lightscattering from slightly rough dielectriclayers

Abstract Reflection of waves from a mirror covered by a random layer of isotropic, absorbing scatterers is studied and the angular distribution of the scattered intensity is calculated both analytically and numerically. It is shown that backscattering enhancement as well as an enhancement of the incoherent signal in the specular direction take place even in the singly scattered random field. The dependence of the retroreflected intensity is shown to be a non-monotonic function of the depth of the mirror, with a maximum at a depth of the order of the scattering mean free path. Possibilities for employing the results obtained to detect buried strong scatterers and to retrieve parameters of the random media are discussed. In particular, it is shown that in the case of strong absorption the reflecting plane manifests itself by the presence of a peak in the retroreflected intensity which is missing from the scattering diagram of a free-standing or an infinitely thick random layer.