238 research outputs found
Statistical Analysis of Composite Spectra
We consider nearest neighbor spacing distributions of composite ensembles of
levels. These are obtained by combining independently unfolded sequences of
levels containing only few levels each. Two problems arise in the spectral
analysis of such data. One problem lies in fitting the nearest neighbor spacing
distribution to the histogram of level spacings obtained from the data. We show
that the method of Bayesian inference is superior to this procedure. The second
problem occurs when one unfolds such short sequences. We show that the
unfolding procedure generically leads to an overestimate of the chaoticity
parameter. This trend is absent in the presence of long-range level
correlations. Thus, composite ensembles of levels from a system with long-range
spectral stiffness yield reliable information about the chaotic behavior of the
system.Comment: 26 pages, 3 figures; v3: changed conclusions, appendix adde
Phenomenological model for symmetry breaking in chaotic system
We assume that the energy spectrum of a chaotic system undergoing symmetry
breaking transitions can be represented as a superposition of independent level
sequences, one increasing on the expense of the others. The relation between
the fractional level densities of the sequences and the symmetry breaking
interaction is deduced by comparing the asymptotic expression of the
level-number variance with the corresponding expression obtained using the
perturbation theory. This relation is supported by a comparison with previous
numerical calculations. The predictions of the model for the
nearest-neighbor-spacing distribution and the spectral rigidity are in
agreement with the results of an acoustic resonance experiment.Comment: accepted for publication in Physical Review
Effect of Unfolding on the Spectral Statistics of Adjacency Matrices of Complex Networks
Random matrix theory is finding an increasing number of applications in the
context of information theory and communication systems, especially in studying
the properties of complex networks. Such properties include short-term and
long-term correlation. We study the spectral fluctuations of the adjacency of
networks using random-matrix theory. We consider the influence of the spectral
unfolding, which is a necessary procedure to remove the secular properties of
the spectrum, on different spectral statistics. We find that, while the spacing
distribution of the eigenvalues shows little sensitivity to the unfolding
method used, the spectral rigidity has greater sensitivity to unfolding.Comment: Complex Adaptive Systems Conference 201
Superstatistical generalisations of Wishart-Laguerre ensembles of random matrices
Using Beck and Cohen's superstatistics, we introduce in a systematic way a family of generalized Wishart–Laguerre ensembles of random matrices with Dyson index β = 1, 2 and 4. The entries of the data matrix are Gaussian random variables whose variances η fluctuate from one sample to another according to a certain probability density f(η) and a single deformation parameter γ. Three superstatistical classes for f(η) are usually considered: χ2-, inverse χ2- and log-normal distributions. While the first class, already considered by two of the authors, leads to a power-law decay of the spectral density, we here introduce and solve exactly a superposition of Wishart–Laguerre ensembles with inverse χ2-distribution. The corresponding macroscopic spectral density is given by a γ-deformation of the semi-circle and Marčenko–Pastur laws, on a non-compact support with exponential tails. After discussing in detail the validity of Wigner's surmise in the Wishart–Laguerre class, we introduce a generalized γ-dependent surmise with stretched-exponential tails, which well approximates the individual level spacing distribution in the bulk. The analytical results are in excellent agreement with numerical simulations. To illustrate our findings we compare the χ2- and inverse χ2-classes to empirical data from financial covariance matrices
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