Matrices coupled in a chain. I. Eigenvalue correlations

The general correlation function for the eigenvalues of $p$ complex hermitian matrices coupled in a chain is given as a single determinant. For this we use a slight generalization of a theorem of Dyson.Comment: ftex eynmeh.tex, 2 files, 8 pages Submitted to: J. Phys.

Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Random matrix ensembles associated with Lax matrices

A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of an integrable structure permits to calculate the joint distribution of eigenvalues for these matrices analytically. Spectral statistics of these ensembles are quite unusual and in many cases give rigorously new examples of intermediate statistics

Finite-difference distributions for the Ginibre ensemble

The Ginibre ensemble of complex random matrices is studied. The complex valued random variable of second difference of complex energy levels is defined. For the N=3 dimensional ensemble are calculated distributions of second difference, of real and imaginary parts of second difference, as well as of its radius and of its argument (angle). For the generic N-dimensional Ginibre ensemble an exact analytical formula for second difference's distribution is derived. The comparison with real valued random variable of second difference of adjacent real valued energy levels for Gaussian orthogonal, unitary, and symplectic, ensemble of random matrices as well as for Poisson ensemble is provided.Comment: 8 pages, a number of small changes in the tex

Generalization of the Poisson kernel to the superconducting random-matrix ensembles

We calculate the distribution of the scattering matrix at the Fermi level for chaotic normal-superconducting systems for the case of arbitrary coupling of the scattering region to the scattering channels. The derivation is based on the assumption of uniformly distributed scattering matrices at ideal coupling, which holds in the absence of a gap in the quasiparticle excitation spectrum. The resulting distribution generalizes the Poisson kernel to the nonstandard symmetry classes introduced by Altland and Zirnbauer. We show that unlike the Poisson kernel, our result cannot be obtained by combining the maximum entropy principle with the analyticity-ergodicity constraint. As a simple application, we calculate the distribution of the conductance for a single-channel chaotic Andreev quantum dot in a magnetic field.Comment: 7 pages, 2 figure

Glassy dynamics in granular compaction

Two models are presented to study the influence of slow dynamics on granular compaction. It is found in both cases that high values of packing fraction are achieved only by the slow relaxation of cooperative structures. Ongoing work to study the full implications of these results is discussed.Comment: 12 pages, 9 figures; accepted in J. Phys: Condensed Matter, proceedings of the Trieste workshop on 'Unifying concepts in glass physics

Two photon annihilation operators and squeezed vacuum

Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than

A 3-component laser-Doppler velocimeter data acquisition and reduction system

A laser doppler velocimeter capable of measuring all three components of velocity simultaneously in low-speed flows is described. All the mean velocities, Reynolds stresses, and higher-order products can be evaluated. The approach followed is to split one of the two colors used in a 2-D system, thus creating a third set of beams which is then focused in the flow from an off-axis direction. The third velocity component is computed from the known geometry of the system. The laser optical hardware and the data acquisition electronics are described in detail. In addition, full operating procedures and listings of the software (written in BASIC and ASSEMBLY languages) are also included. Some typical measurements obtained with this system in a vortex/mixing layer interaction are presented and compared directly to those obtained with a cross-wire system

A Theory of Errors in Quantum Measurement

It is common to model random errors in a classical measurement by the normal (Gaussian) distribution, because of the central limit theorem. In the quantum theory, the analogous hypothesis is that the matrix elements of the error in an observable are distributed normally. We obtain the probability distribution this implies for the outcome of a measurement, exactly for the case of 2x2 matrices and in the steepest descent approximation in general. Due to the phenomenon of `level repulsion', the probability distributions obtained are quite different from the Gaussian.Comment: Based on talk at "Spacetime and Fundamental Interactions: Quantum Aspects" A conference to honor A. P. Balachandran's 65th Birthda

Signatures of Random Matrix Theory in the Discrete Energy Spectra of Subnanosize Metallic Clusters

Lead clusters deposited on Si(111) substrates have been studied at low temperatures using scanning tunneling microscopy and spectroscopy. The current-voltage characteristics exhibit current peaks that are irregularly spaced and varied in height. The statistics of the distribution of peak heights and spacings are in agreement with random matrix theory for several clusters. The distributions have also been studied as a function of cluster shape.Comment: 10 pages, 9 figures, to appear in Phys. Rev.