Predictions Of Random Matrix Theory Research Articles

Collective behavior of cryptocurrency price changes

Digital assets termed cryptocurrencies are correlated. We analyze cross correlations between price changes of different cryptocurrencies using methods of random matrix theory and minimum spanning trees. We find that the cross correlation matrix exhibits non-trivial hierarchical structures and groupings of cryptocurrency pairs, which are not present in the partial cross correlations. In sharp contrast to the predictions for other financial markets, we discover that most of the eigenvalues in the spectrum of the cross correlation matrix do not agree with the universal predictions of random matrix theory, but the few of the largest eigenvalues deviate as expected. The minimum spanning tree of cryptocurrency cross correlations reveals distinct community structures that are surprisingly stable. Collective behaviors that are present in the cryptocurrency market can be useful for the construction of portfolio of cryptocurrencies as well as for future research on the subject.

Experimental evidence of chaos in the bound states of 208Pb

Theoretical calculations, especially shell-model calculations, have shown a strongly chaotic behavior of bound states at higher excitation energy, in regions of high level density. However, it had not been possible up to now to observe chaos in the experimental bound energy levels of any single nucleus. In this paper we study the spectral fluctuations of the 208Pb nucleus using the complete experimental spectrum of 151 states up to excitation energies of 6.20 MeV. For natural parity states the results are very close to the predictions of Random Matrix Theory (RMT) for the nearest-neighbor spacing distribution. By contrast, the results for unnatural parity states are far from RMT behavior. We interpret these results as a consequence of the strength of the residual interaction in 208Pb, which, according to experimental data, is much stronger for natural than for unnatural parity states. In addition our results show that chaotic and non-chaotic nuclear states coexist in the same energy region of the spectrum.

Universality in the dynamical properties of seismic vibrations

We have studied the statistical properties of the observed magnitudes of seismic vibration data in discrete time in an attempt to understand the underlying complex dynamical processes. The observed magnitude data are taken from six different geographical locations. All possible magnitudes are considered in the analysis including catastrophic vibrations, foreshocks, aftershocks and commonplace daily vibrations. The probability distribution functions of these data sets obey scaling law and display a certain universality characteristic. To investigate the universality features in the observed data generated by a complex process, we applied Random Matrix Theory (RMT) in the framework of Gaussian Orthogonal Ensemble (GOE). For all these six places the observed data show a close fit with the predictions of RMT. This reinforces the idea of universality in the dynamical processes generating seismic vibrations.

Polarizability of electrically induced magnetic vortex plasma

Electric field control of magnetic structures, particularly topological defects in magnetoelectric materials, draws a great attention in recent years, which has led to experimental success in creation and manipulation by electric field of single magnetic defects, such as domain walls and skyrmions. In this work we explore a scenario of electric field creation of another type of topological defects -- magnetic vortices and antivortices, which are characteristic for materials with easy plane (XY) symmetry. Each magnetic (anti)vortex in magnetoelectric materials (such as type-II multiferroics) possesses a quantized magnetic and an electric charge, where the former is responsible for interaction between vortices and the latter couples the vortices to electric field. This property of magnetic vortices opens a peculiar possibility of creation of magnetic vortex plasma by non-uniform electric fields. We show that the electric field, created by a cantilever tip, produces a "magnetic atom" with a localized spatially ordered spot of vortices ("nucleus" of the atom) surrounded by antivortices ("electronic shells"). We analytically find the vortex density distribution profile and temperature dependence of polarizability of this structure and confirm it numerically. We show that electric polarizability of the "magnetic atom" depends on temperature as $\alpha \sim 1/T^{1-\eta}$ ($\eta>0$), which is consistent with Euclidean random matrix theory prediction.

An electric field induced delocalization transition in second-harmonic generation effect

The emergence of second-harmonic generation (SHG) is a pivotal issue to the development of nano-optical devices and interfaces. Here, we perform a quantum analysis of the spectral statistics of the SHG resonances with the aim of determining the instability of this process. We compute different independent statistical properties to being chaotic. We find that the short-range fluctuations of the resonances at electric fields with different intensities are in agreement with random matrix theory (RMT) predictions. We conclude that although in low intensities of electric field the SHG dynamics follows Poisson statistics, in high intensities the dynamics is chaotic (ergodic). On the other hand, a study of the singularity spectrum of eigenstates provides additional evidence supporting RMT predictions. We also analyze the inverse participation ratio of eigenstates and find that it equals \(\sim\)0.5 in localized phase, and approaches to \(1/dim(\mathcal {H})\) in delocalized (chaotic) phase.

Examination of experimental evidence of chaos in the bound states of Pb208

We study the spectral fluctuations of the $^{208}$Pb nucleus using the complete experimental spectrum of 151 states up to excitation energies of $6.20$ MeV recently identified at the Maier-Leibnitz-Laboratorium at Garching, Germany. For natural parity states the results are very close to the predictions of Random Matrix Theory (RMT) for the nearest-neighbor spacing distribution. A quantitative estimate of the agreement is given by the Brody parameter $\omega$, which takes the value $\omega=0$ for regular systems and $\omega \simeq 1$ for chaotic systems. We obtain $\omega=0.85 \pm 0.02$ which is, to our knowledge, the closest value to chaos ever observed in experimental bound states of nuclei. By contrast, the results for unnatural parity states are far from RMT behavior. We interpret these results as a consequence of the strength of the residual interaction in $^{208}$Pb, which, according to experimental data, is much stronger for natural than for unnatural parity states. In addition our results show that chaotic and non-chaotic nuclear states coexist in the same energy region of the spectrum.

Vibrational energy transport in molecules and the statistical properties of vibrational modes

Statistical properties of the eigenmodes computed for two molecules, dodecane and perfluorododecane, are examined and compared with predictions of random matrix theory. The eigenmode statistics of the heat carrying modes of perfluorododecane correspond to Porter-Thomas statistics, whereas those for dodecane do not. Vibrational energy transport in the two molecules is also computed and found to be diffusive in perfluorododecane but not in dodecane, consistent with recent experiments. The correspondence between eigenmode statistics and vibrational energy transport dynamics in molecules as well as thermalization in molecules are discussed.

The spatiotemporal dynamic analysis of the implied market information and characteristics of the correlation coefficient matrix of the international crude oil price returns

The international crude oil market plays a crucial role in economies, and the studies of the correlation, risk and synchronization of the international crude oil market have important implications for the security and stability of the country, avoidance of business risk and people’s daily lives. We investigate the information and characteristics of the international crude oil market (1999–2015) based on the random matrix theory (RMT). Firstly, we identify richer information in the largest eigenvalues deviating from RMT predictions for the international crude oil market; the international crude oil market can be roughly divided into ten different periods by the methods of eigenvectors and characteristic combination, and the implied market information of the correlation coefficient matrix is advanced. Secondly, we study the characteristics of the international crude oil market by the methods of system risk entropy, dynamic synchronous ratio, dynamic non-synchronous ratio and dynamic clustering algorithm. The results show that the international crude oil market is full of risk. The synchronization of the international crude oil market is very strong, and WTI and Brent occupy a very important position in the international crude oil market.

Statistical distribution of components of energy eigenfunctions: from nearly-integrable to chaotic

We study the statistical distribution of components in the non-perturbative parts of energy eigenfunctions (EFs), in which main bodies of the EFs lie. Our numerical simulations in five models show that deviation of the distribution from the prediction of random matrix theory (RMT) is useful in characterizing the process from nearly-integrable to chaotic, in a way somewhat similar to the nearest-level-spacing distribution. But, the statistics of EFs reveals some more properties, as described below. (i) In the process of approaching quantum chaos, the distribution of components shows a delay feature compared with the nearest-level-spacing distribution in most of the models studied. (ii) In the quantum chaotic regime, the distribution of components always shows small but notable deviation from the prediction of RMT in models possessing classical counterparts, while, the deviation can be almost negligible in models not possessing classical counterparts. (iii) In models whose Hamiltonian matrices possess a clear band structure, tails of EFs show statistical behaviors obviously different from those in the main bodies, while, the difference is smaller for Hamiltonian matrices without a clear band structure.

Approach to chaos in ultracold atomic and molecular physics: Statistics of near-threshold bound states for Li+CaH and Li+CaF

We calculate near-threshold bound states and Feshbach resonance positions for atom + rigid-rotor models of the highly anisotropic systems Li+CaH and Li+CaF. We perform statistical analysis on the resonance positions to compare with the predictions of random matrix theory. For Li+CaH with total angular momentum $J=0$ we find fully chaotic behavior in both the nearest-neighbor spacing distribution and the level number variance. However, for $J>0$ we find different behavior due to the presence of a nearly conserved quantum number. Li+CaF ($J=0$) also shows apparently reduced levels of chaotic behavior despite its stronger effective coupling. We suggest this may indicate the development of another good quantum number relating to a bending motion of the complex. However, continuously varying the rotational constant over a wide range shows unexpected structure in the degree of chaotic behavior, including a dramatic reduction around the rotational constant of CaF. This demonstrates the complexity of the relationship between coupling and chaotic behavior.

Localization in chaotic systems with a single-channel opening.

We introduce a single-channel opening in a random Hamiltonian and a quantized chaotic map: localization on the opening occurs as a sensible deviation of the wave-function statistics from the predictions of random matrix theory, even in the semiclassical limit. Increasing the coupling to the open channel in the quantum model, we observe a similar picture to resonance trapping, made of a few fast-decaying states, whose left (right) eigenfunctions are entirely localized on the (preimage of the) opening, and plentiful long-lived states, whose probability density is instead suppressed at the opening. For the latter, we derive and test a linear relation between the wave-function intensities and the decay rates, similar to the Breit-Wigner law. We then analyze the statistics of the eigenfunctions of the corresponding (discretized) classical propagator, finding a similar behavior to the quantum system only in the weak-coupling regime.

Universality in spectral statistics of open quantum graphs.

The quantum evolution maps of closed chaotic quantum graphs are unitary and known to have universal spectral correlations matching predictions of random matrix theory. In chaotic graphs with absorption the quantum maps become nonunitary. We show that their spectral statistics exhibit universality at the soft edges of the spectrum. The same spectral behavior is observed in many classical nonunitary ensembles of random matrices with rotationally invariant measures.

Statistics of time delay and scattering correlation functions in chaotic systems. II. Semiclassical approximation

We consider S-matrix correlation functions for a chaotic cavity having M open channels, in the absence of time-reversal invariance. Relying on a semiclassical approximation, we compute the average over E of the quantities Tr[S†(E − ϵ) S(E + ϵ)]n, for general positive integer n. Our result is an infinite series in ϵ, whose coefficients are rational functions of M. From this, we extract moments of the time delay matrix Q = − iħS†dS/dE and check that the first 8 of them agree with the random matrix theory prediction from our previous paper [M. Novaes, J. Math. Phys. 56, 062110 (2015)].

Analysing correlations after the financial crisis of 2008 and multifractality in global financial time series

We apply random matrix theory (RMT) to investigate the structure of cross-correlation in 20 global financial time series after the global financial crisis of 2008. We find that the largest eigenvalue deviates from the RMT prediction and is sensitive to the financial crisis. We find that the components of eigenvectors corresponding to the second largest eigenvalue changes sign in response to the crisis. We show that 20 global financial indices exhibit multifractality. We find that the origin of multifractality is due to the long-range correlations as well as broad probability function in the financial indices, with the exception of the index of Taiwan, as in all other indices the multifractal degree for shuffled and surrogate series is weaker than the original series. We fit the binomial multifractal model to the global financial indices.

A Big Data Architecture Design for Smart Grids Based on Random Matrix Theory

Model-based analysis tools, built on assumptions and simplifications, are difficult to handle smart grids with data characterized by volume, velocity, variety, and veracity (i.e., 4Vs data). This paper, using random matrix theory (RMT), motivates data-driven tools to perceive the complex grids in high-dimension; meanwhile, an architecture with detailed procedures is proposed. In algorithm perspective, the architecture performs a high-dimensional analysis and compares the findings with RMT predictions to conduct anomaly detections. Mean spectral radius (MSR), as a statistical indicator, is defined to reflect the correlations of system data in different dimensions. In management mode perspective, a group-work mode is discussed for smart grids operation. This mode breaks through regional limitations for energy flows and data flows, and makes advanced big data analyses possible. For a specific large-scale zone-dividing system with multiple connected utilities, each site, operating under the group-work mode, is able to work out the regional MSR only with its own measured/simulated data. The large-scale interconnected system, in this way, is naturally decoupled from statistical parameters perspective, rather than from engineering models perspective. Furthermore, a comparative analysis of these distributed MSRs, even with imperceptible different raw data, will produce a contour line to detect the event and locate the source. It demonstrates that the architecture is compatible with the block calculation only using the regional small database; beyond that, this architecture, as a data-driven solution, is sensitive to system situation awareness, and practical for real large-scale interconnected systems. Five case studies and their visualizations validate the designed architecture in various fields of power systems. To our best knowledge, this paper is the first attempt to apply big data technology into smart grids.

On the concentration of large deviations for fat tailed distributions, with application to financial data

Large deviations for fat tailed distributions, i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable. The regime of large deviations is separated from the regime of typical fluctuations by a phase transition where the symmetry between the points in the sample is spontaneously broken. For stochastic processes with a fat tailed microscopic noise, this implies that, while typical realizations are well described by a diffusion process with continuous sample paths, large deviation paths are typically discontinuous. For eigenvalues of random matrices with fat tailed distributed elements, a large deviation where the trace of the matrix is anomalously large concentrates on just a single eigenvalue, whereas in the thin tailed world the large deviation affects the whole distribution.These results find a natural application to finance. Since the price dynamics of financial stocks are characterized by fat tailed increments, large fluctuations in stock prices are expected to be realized by discrete jumps. Interestingly, we find that large excursions of prices are more likely realized by continuous drifts rather than by discontinuous jumps. Indeed, auto correlations suppress the concentration of large deviations. Financial covariance matrices also exhibit an anomalously large eigenvalue, the market mode, as compared to the prediction of random matrix theory. We show that this is explained by a large deviation with excess covariance rather than by one with excess volatility.

Open problems in applying random-matrix theory to nuclear reactions

Problems in applying random-matrix theory (RMT) to nuclear reactions arise in two domains. To justify the approach, statistical properties of isolated resonances observed experimentally must agree with RMT predictions. That agreement is less striking than would be desirable. In the implementation of the approach, the range of theoretically predicted observables is too narrow.

Systemic risk and spatiotemporal dynamics of the US housing market

Housing markets play a crucial role in economies and the collapse of a real-estate bubble usually destabilizes the financial system and causes economic recessions. We investigate the systemic risk and spatiotemporal dynamics of the US housing market (1975–2011) at the state level based on the Random Matrix Theory (RMT). We identify richer economic information in the largest eigenvalues deviating from RMT predictions for the housing market than for stock markets and find that the component signs of the eigenvectors contain either geographical information or the extent of differences in house price growth rates or both. By looking at the evolution of different quantities such as eigenvalues and eigenvectors, we find that the US housing market experienced six different regimes, which is consistent with the evolution of state clusters identified by the box clustering algorithm and the consensus clustering algorithm on the partial correlation matrices. We find that dramatic increases in the systemic risk are usually accompanied by regime shifts, which provide a means of early detection of housing bubbles.

Spectral properties of a two dimensional photonic crystal with quasi-integrable geometry

In this paper we study the statistical properties of the allowed frequencies for electromagnetic waves propagating in two-dimensional photonic crystals with quasi-integrable geometry. We compute the level spacing, group velocity, and curvature distributions (P(s), P(v), and P(c), respectively) and compare them with the corresponding random matrix theory predictions. Due to the quasi-integrability of the crystal we observe signatures of intermediate statistics in P(s) and P(c) for high refractive index contrasts.

Impedance and Scattering Variance Ratios of Complicated Wave Scattering Systems in the Low Loss Regime

Random matrix theory (RMT) successfully predicts universal statistical properties of complicated wave scattering systems in the semiclassical limit, while the random coupling model offers a complete statistical model with a simple additive formula in terms of impedance to combine the predictions of RMT and nonuniversal system-specific features. The statistics of measured wave properties generally have nonuniversal features. However, ratios of the variances of elements of the impedance matrix are predicted to be independent of such nonuniversal features and thus should be universal functions of the overall system loss. In contrast with impedance variance ratios, scattering variance ratios depends on nonuniversal features unless the system is in the high loss regime. In this paper, we present numerical tests of the predicted universal impedance variance ratios and show that an insufficient sample size can lead to apparent deviation from the theory, particularly in the low loss regime. Experimental tests are carried out in three two-port microwave cavities with varied loss parameters, including a novel experimental system with a superconducting microwave billiard, to test the variance-ratio predictions in the low loss time-reversal-invariant regime. It is found that the experimental results agree with the theoretical predictions to the extent permitted by the finite sample size.