Results In Random Matrix Theory Research Articles

Statistical properties of wave transport through surface-disordered waveguides

We review some general statistical properties of wave transport through surface disordered waveguides. These systems are shown to present both striking similarities and differences with respect to quasi-one-dimensional waveguides with volume disorder. The statistical properties are analysed using extensive numerical calculations and random matrix theory results. The transport properties are characterized by the statistical behaviour of different transport coefficients that can be defined for both classical (light, microwaves, sound, etc.) and quantum (electrons) waves. In analogy with bulk-disordered systems, the behaviour of the waveguide conductance/resistance (defined for both classical and quantum waves) as a function of the system length defines three different transport regimes: ballistic, diffusive and localization. However, the coupling between waveguide modes presents significant differences with respect to the coupling induced by volume defects. For any incoming mode, there is a strong preference for the forward propagation through the lowest mode. For narrow waveguides, the statistics of reflection coefficients (reflected speckle pattern) present strong finite-size effects which can be surprisingly well described by random matrix theory. Special attention is paid to the fundamental problem of the transition between different regimes. The long-standing problems of the phase randomization process between ballistic and diffusive regimes and the evolution of the conductance statistical distribution in the transition from diffusion (Gaussian statistics) to localization (log normal statistics) are also discussed.

Quantifying economic fluctuations

This manuscript is a brief summary of a talk designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena—scale invariance and universality—can be useful in guiding research on interpreting empirical data on economic fluctuations. Using this conceptual framework as a guide, we empirically quantify the relation between trading activity—measured by the number of transactions N—and the price change G( t) for a given stock, over a time interval [ t, t+Δ t]. We relate the time-dependent standard deviation of price changes—volatility—to two microscopic quantities: the number of transactions N( t) in Δ t and the variance W 2( t) of the price changes for all transactions in Δ t. We find that the long-ranged volatility correlations are largely due to those of N. We then argue that the tail-exponent of the distribution of N is insufficient to account for the tail-exponent of P{ G> x}. Since N and W display only weak inter-dependency, our results show that the fat tails of the distribution P{ G> x} arises from W. Finally, we review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory (RMT). RMT makes predictions for “universal” properties that do not depend on the interactions between the elements comprising the system, and deviations from RMT provide clues regarding system-specific properties. We compare the statistics of the cross-correlation matrix C —whose elements C ij are the correlation coefficients of price fluctuations of stock i and j—against a random matrix having the same symmetry properties. It is found that RMT methods can distinguish random and non-random parts of C . The non-random part of C which deviates from RMT results, provides information regarding genuine collective behavior among stocks. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behavior of the response function at the critical point (zero magnetic field) leads to large fluctuations, and we discuss a curious “symmetry breaking”, a feature qualitatively identical to the behavior of the probability density of the magnetization for fixed values of the inverse temperature.

Collective behavior of stock price movements—a random matrix theory approach

We review recent work on quantifying collective behavior among stocks by applying the conceptual framework of random matrix theory (RMT), developed in physics to describe the energy levels of complex systems. RMT makes predictions for “universal” properties that do not depend on the interactions between the elements comprising the system; deviations from RMT provide clues regarding system-specific properties. We compare the statistics of the cross-correlation matrix C —whose elements C ij are the correlation coefficients of price fluctuations of stock i and j—against a random matrix having the same symmetry properties. It is found that RMT methods can distinguish random and non-random parts of C . The non-random part of C which deviates from RMT results, provides information regarding genuine collective behavior among stocks.

Beyond the Thouless energy

The distribution and the correlations of the small eigenvalues of the Dirac operator are described by random matrix theory (RMT) up to the Thouless energy E c ∝ 1 √V , where V is the physical volume. For somewhat larger energies, the same quantities can be described by chiral perturbation theory (chPT). For most quantities there is an intermediate energy regime, roughly 1 V < E < 1 √V , where the results of RMT and chPT agree with each other. We test these predictions by constructing the connected and disconnected scalar susceptibilities from Dirac spectra obtained in quenched SU(2) and SU(3) simulations with staggered fermions for a variety of lattice sizes and coupling constants. In deriving the predictions of chPT, it is important to take into account only those symmetries which are exactly realized on the lattice.

Distribution of level curvatures for the Anderson model at the localization-delocalization transition.

We compute the distribution function of single-level curvatures, $P(k)$, for a tight binding model with site disorder, on a cubic lattice. In metals $P(k)$ is very close to the predictions of the random-matrix theory (RMT). In insulators $P(k)$ has a logarithmically-normal form. At the Anderson localization-delocalization transition $P(k)$ fits very well the proposed novel distribution $P(k)\propto (1+k^{\mu})^{3/\mu}$ with $\mu \approx 1.58$, which approaches the RMT result for large $k$ and is non-analytical at small $k$. We ascribe such a non-analiticity to the spatial multifractality of the critical wave functions.

Fokker-Planck description of the transfer-matrix limiting distribution in the scattering approach to quantum transport.

The scattering approach to quantum transport through a disordered quasi-one-dimensional conductor in the insulating regime is discussed in terms of its transfer matrix $\bbox{T}$. A model of $N$ one-dimensional wires which are coupled by random hopping matrix elements is compared with the transfer matrix model of Mello and Tomsovic. We derive and discuss the complete Fokker-Planck equation which describes the evolution of the probability distribution of $\bbox{TT}^{\dagger}$ with system length in the insulating regime. It is demonstrated that the eigenvalues of $\ln\bbox{TT}^{\dagger}$ have a multivariate Gaussian limiting probability distribution. The parameters of the distribution are expressed in terms of averages over the stationary distribution of the eigenvectors of $\bbox{TT}^{\dagger}$. We compare the general form of the limiting distribution with results of random matrix theory and the Dorokhov-Mello-Pereyra-Kumar equation.

Quantum chaos in collinear (He, H2+) collisions

The quasibound spectrum of the transition state in collinear (He, H2+) collisions is obtained from time-dependent wave packet calculations. Examination of short- and long-range correlations in the eigenvalue spectra through a study of the nearest neighbor spacing distribution, P(s), and the spectral rigidity, Δ3(L), reveals signatures of quantum chaotic behavior. Analysis in the time domain is carried out by computing the survival probability 〈〈P(t)〉〉 averaged over initial states and Hamiltonian. All these indicators show intermediate behavior between regular and chaotic. A quantitative comparison of 〈〈P(t)〉〉 with the results of random matrix theory provides an estimate of the fraction of phase space exhibiting chaotic behavior, in reasonable agreement with the classical dynamics. We also analyse the dynamical evolution of coherent Gaussian wave packets located initially in different regions of phase space and compute the survival probability, power spectrum and the volume of phase space over which the wave packet spreads and illustrate the different behaviors.

Effective plasma model for the level correlations at the mobility edge

We consider the mapping of the energy level statistics for a d-dimensional disordered electron system at the mobility edge between metallic and insulating phases onto the model of a classical one-dimensional 'plasma' of fictitious particles. We deduce the effective pairwise interaction in the plasma that is consistent with the known universal two-level correlation function at the mobility edge and show that for level separation epsilon >> Delta it decreases as ( Delta / epsilon )gamma where Delta is the mean-level spacing, and gamma is the critical exponent related to the known critical exponent v of the correlation length as gamma =1-(vd)-1. We apply the plasma model to generalize Wigner's semicircle law, and to derive the large-energy asymptotic form of the nearest-level distribution. In the limit gamma to 0, which corresponds to the original Dyson mapping onto the plasma with logarithmic repulsion, we recover the classical results of Wigner-Dyson random matrix theory.

Nonexponential decay of a stochastic one-channel system.

A general formula is presented that expresses the temporal evolution of compound systems. It connects the time dependence of the density matrix with the energy dependence of the scattering matrix. Using results of random matrix theory, we then study the decay behavior of stochastic compound systems with arbitrary coupling between bound states and decay channels. As an example we consider the case of one open channel and prove a nonexponential decay law for all times that asymptotically is of the form ${\mathit{t}}^{\mathrm{\ensuremath{-}}3/2}$.

Regular mechanism of parity and time invariance nonconserving effects enhancement in neutron capture and scattering near p-wave compound resonances.

Recent measurements of parity nonconserving (PNC) effects in $^{238}\mathrm{U}$ and $^{232}\mathrm{Th}$ contradict the results of random matrix theory for nuclear compound states. In this work the value of the PNC effect is expressed in terms of a wave function at the nuclear surface where the wave function of the compound state is not ``random'' due to boundary conditions. A mechanism is suggested which can explain the permanent sign and large value of these effects. The correlations between compound-state components are considered. The T- and P-odd effects can be expressed in terms of observed P-odd effects. The ``dynamical enhancement'' of small interactions in other reactions and other systems is also discussed.

Random matrix theory in semiclassical quantum mechanics of chaotic systems

The statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles. The author explains this correspondence, including the fact that long-ranged spectral statistics show deviations from the results of random matrix theory. The method depends on the ambiguity of quantisation of a given classical system; although the energy levels depend on the particular quantisation used, the spectral statistics are assumed to depend only on the classical motion. The ambiguity of quantisation can be represented by a small perturbation acting on the Hamiltonian. This perturbation executes a random walk, which causes its matrix elements to undergo a random walk. If the matrix elements are completely uncorrelated, the spectral statistics are those of random matrix theory. In the case of classically chaotic systems, the matrix elements are constrained by sum rules related to classical periodic orbits. This leads to deviations of the long-ranged spectral statistics from the predictions of random matrix theory.

Quantum chaos and fluctuation properties of regular and irregular spectra

Fluctuation properties of the regular and irregular energy levels for the Henon-Heiles Hamiltonian are examined. The spacing distributions and the calculated values of the Δ3-statistic show that there is no difference in the short range correlation properties of these spectra. Remarkably, the Δ3 values agree with the results of random matrix theory.

Random matrix theory and the statistical mechanics of disordered systems

A new method, that systematically combines results of random matrix theory and the usual statistical mechanics, is described to study thermodynamic properties of disordered systems. Two exactly solvable models are examined in this formulation to illustrate the usefulness of this method for systems described by random as well as non-random Hamiltonian.

Influence of the statistical distribution of level parameters on compound nuclear cross sections for strongly overlapping resonances

We examine numerically the influence of the distribution of the poles and residues of the $K$ matrix on the elastic enhancement factor $W$ in the Hauser-Feshbach formula. For a distribution of pole parameters consistent with the results of random matrix theory, our results strongly suggest a value of $W=2.0$, in the limit of strongly overlapping resonances.NUCLEAR REACTIONS Influence of statistical distribution of level parameters on compound nucleus reaction cross section for strong absorption.

Level-density fluctuations and two-body versus multi-body interactions

The level density, as calculated from many-particle spectroscopic matrices, is sensitive to the p-body nature of the interaction, changing from an approximately Gaussian form for 2-body interactions to the Wigner semi-circular form when all the particles interact simultaneously (as happens in the random-matrix realization of the matrices). The difference, non-local in energy, is however not usable for investigating the “particle rank”, p, of the interaction, since in actual nuclei the restricted density cannot be realized over a wide enough energy range to define its non-local behavior. One is then led to ask whether the (local) level-density fluctuations, which are observed mainly in slow-neutron spectroscopy and which are usable in searching for hidden symmetries, can be used also in deciding whether the interaction is predominantly 2-body or multi-body. The fluctuations produced by Gaussian orthogonal ensembles (GOE) of finite-dimensional matrices (dimensionalities 50 and 294) have been compared with those produced by corresponding ensembles defined by 2-body interactions, as well as with results of recent very large shell-model calculations (dimensionalities 500–1200) and recent experiments. Comparison is made by means of the spacing distributions of various orders, the nearest-neighbor-spacing covariance, and the long- and short-range-order statistics of Dyson and Mehta. The fluctuations are found to be essentially identical for all cases, and to agree with theoretical GOE results where such are available. It appears then that the results of conventional random-matrix theory apply under very general conditions, but the physical information which is carried by the level-density fluctuations alone seems to be quite meager.