Parametric modelling of Poisson-Gaussian random matrix ensembles

We analyse a scheme of transition from the Poissonian statistics for quantum levels to the Gaussian one of random matrix ensembles in the framework of level dynamics discussed by Yukawa. We propose a means of connecting these two limiting statistics by showing a result that Yukawa's parameter / of the exponential family can be efficiently replaced by the ratio <E>/<Q> which reflects directly a degree of the eigenvalue correlations of each sample matrix in the ensemble. On this basis, we discuss a correspondence between the level statistics of a generic quantum system and its classical regular/chaotic dynamics in terms of the semiclassical power spectrum and its second moment formulated by Feingold-Peres and Prosen-Robnik. We also discuss some limiting proceduresN (infinite limit of the matrix dimension) pertinent to the Gaussian ensembles, and remark about the possibility offractional power law of Brody's type.     

Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

Eigenfunction entropy and spectral compressibility for critical random matrix ensembles

Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D(1) and the spectral compressibility χ are related by the simple equation χ+D(1)/d=1, where d is system dimensionality.     

Power series for level spacing functions of random matrix ensembles

A remarkable set of identities among (odd and even) spheroidal functions and their eigenvalues is indicated. This is then used to derive explicitly the general term in the power series expansion ofE (r, s), the probability that a randomly chosen interval of lengths contains exactlyr levels. The parameter takes the values 1, 2 or 4 according as the ensemble considered is orthogonal, unitary or symplectic.Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique     

One- plus two-body random matrix ensembles with spin: Results for pairing correlations

For EGOE(1+2)-s ensemble for fermions, in the strong coupling region, partial densities over pairing subspaces follow Gaussian form and propagation formulas for their centroids and variances are derived. Similarly for this ensemble: (i) pair transfer strength sums, a statistic for chaos, are shown to follow a simple form; (ii) a quantity used in conductance peak spacings analysis is shown to exhibit bimodal form when pairing is stronger than the exchange interaction.     

Energy absorption in time-dependent unitary random matrix ensembles: Dynamic versus anderson localization

JETP Letters - We consider energy absorption in an externally driven complex system of noninteracting fermions with the chaotic underlying dynamics described by the unitary random matrices. In the...     

Level density fluctuations and random matrix theory

The main purpose of this work is to elucidate whether there are significant differences in the local fluctuation properties between two-body (TBRE) and orthogonal (OE) ensembles of random matrices. Emphasis is put on the validity of ergodic properties, and results obtained by numerical means are discussed from that point of view. Spectral and ensemble averaging procedures are compared. All the local properties studied show compatibility between TBRE and OE results, and no significant evidence of inconsistency of theoretical predictions and experimental data is found.     

Centrality in High-Energy Nuclear Collisions and Random Matrix Theory.

I discuss the results from a study of the central ^12CC collisions at 4.2 A GeV/c. The data have been analyzed using a new method based on the Random Matrix Theory. The simulation data coming from the Ultra Relativistic Quantum Molecular Dynamics code were used in the analyses. I found that the behavior of the nearest neighbor spacing distribution for the protons, neutrons and neutral pions depends critically on the multiplicity of secondary particles for simulated data. I conclude that the obtained results offer the possibility of fixing the centrality using the critical values of the multiplicity.     

Centrality of the collision and random matrix theory

I discuss the results from a study of the central 12CC collisions at 4.2 A GeV/c.The data have been analyzed using a new method based on the Random Matrix Theory.The simulation data coming from the Ultra Relativistic Quantum Molecular Dynamics code were used in the analyses.I found that the behavior of the nearest neighbor spacing distribution for the protons,neutrons and neutral pions depends critically on the multiplicity of secondary particles for simulated data.I conclude that the obtained results offer the possibility of fixing the centrality using the critical values of the multiplicity.     

Universal wide correlators in non-gaussian orthogonal,unitary and symplectic random matrix ensembles

We calculate wide distance connected correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles by solving the loop equation in the 1/N expansion. The multilevel correlator is shown to be universal in the large-N limit. We show the algorithm to obtain the connected correlator to an arbitrary order in the 1/N expansion.     

Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems

Universal properties of simple quantum systems whose classical counter parts are chaotic, are modeled by the classical random matrix ensembles and their interpolations/deformations. However for finite interacting many-particle systems such as atoms, molecules, nuclei and mesoscopic systems (atomic clusters, helium droplets, quantum dots, etc.) for wider range of phenomena, it is essential to include information such as particle number, number of single-particle orbits, lower particle rank of the interaction, etc. These considerations led to resurgence of interest in investigating in detail the so-called embedded random matrix ensembles and their various deformed versions. Besides giving a overview of the basic results of embedded ensembles for the smoothed state densities and transition matrix elements, recent progress in investigating these ensembles with various deformations, for deriving a statistical mechanics (with relationships between quantum chaos, thermalization, phase transitions and Fock space localization, etc.) for isolated finite systems with few particles is briefly discussed. These results constitute new progress in deriving a basis for statistical spectroscopy (introduced and applied in nuclear structure physics and more recently in atomic physics) and its domains of applicability.     

A novel image encryption scheme based on Kepler's third law and random Hadamard transform

In this paper, a novel image encryption scheme based on Kepler's third law and random Hadamard transform is proposed to ensure the security of a digital image. First, a set of Kepler periodic sequences is generated to permutate image data, which is characteristic of the plain-image and the Kepler's third law. Then, a random Hadamard matrix is constructed by combining the standard Hadamard matrix with the hyper-Chen chaotic system, which is used to further scramble the image coefficients when the image is transformed through random Hadamard transform. In the end, the permuted image presents interweaving diffusion based on two special matrices, which are constructed by Kepler periodic sequence and chaos system. The experimental results and performance analysis show that the proposed encrypted scheme is highly sensitive to the plain-image and external keys, and has a high security and speed, which are very suitable for secure real-time communication of image data.     

Law of addition in random matrix theory

We discuss the problem of adding random matrices, which enables us to study Hamiltonians consisting of a deterministic term plus a random term. Using a diagrammatic approach and introducing the concept of “gluon connectedness”, we calculate the density of energy levels for a wide class of probability distributions governing the random term, thus generalizing a result obtained recently by Brézin, Hikami and Zee. The method used here may be applied to a broad class of problems involving random matrices.     

Compression scheme of sub-images using Karhunen-Loeve transform in three-dimensional integral imaging

In this paper, a compression scheme of sub-image-transformed elemental images using Karhunen-Loeve transform (KLT) in three-dimensional integral imaging is proposed. The proposed scheme provides improved compression efficiency by improving the similarity between elemental images using sub-image transformation. To test the proposed scheme, various elemental images of 3D objects are picked up and the compression process is carried out using KLT. From the experimental results, it is showed that the proposed compression scheme gives us an improved efficiency of 26% as compared with the conventional compression method.     

Chiral random matrix theory and effective theories of QCD

The correlations of the QCD Dirac eigenvalues are studied with use of an extended chiral random matrix model. The inclusion of spatial dependence which the orginal model lacks enables us to investigate the effects of diffusion modes. We get analytical expressions of level correlation functions with non-universal behavior caused by diffusion modes which is characterized by Thouless energy. Pion mode is shown to be responsible for these diffusion effects when QCD vacuum is considered a disorderd medium.     

Renormalizing rectangles and other topics in random matrix theory

We consider random Hermitian matrices made of complex or realM×N rectangular blocks, where the blocks are drawn from various ensembles. These matrices haveN pairs of opposite real nonvanishing eigenvalues, as well asM–N zero eigenvalues (forM>N). These zero eigenvalues are kinematical in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the rectangularityr=M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical -function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r–1| is held fixed asN, theN nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. Asr1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal thatr1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.     

Correlation and volatility in an Indian stock market: A random matrix approach

We examine the volatility of an Indian stock market in terms of correlation of stocks and quantify the volatility using the random matrix approach. First we discuss trends observed in the pattern of stock prices in the Bombay Stock Exchange for the three-year period 2000–2002. Random matrix analysis is then applied to study the relationship between the coupling of stocks and volatility. The study uses daily returns of 70 stocks for successive time windows of length 85 days for the year 2001. We compare the properties of matrix C of correlations between price fluctuations in time regimes characterized by different volatilities. Our analyses reveal that (i) the largest (deviating) eigenvalue of C correlates highly with the volatility of the index, (ii) there is a shift in the distribution of the components of the eigenvector corresponding to the largest eigenvalue across regimes of different volatilities, (iii) the inverse participation ratio for this eigenvector anti-correlates significantly with the market fluctuations and finally, (iv) this eigenvector of C can be used to set up a Correlation Index, CI whose temporal evolution is significantly correlated with the volatility of the overall market index.     

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n ^–1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.