Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices

Abul-Magd AY, Akemann G, Vivo P (2008)
Journal of Physics, A: Mathematical and Theoretical 42(17).

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Abstract
Using Beck and Cohen's superstatistics, we introduce in a systematic way afamily of generalised Wishart-Laguerre ensembles of random matrices with Dysonindex $\beta$ = 1,2, and 4. The entries of the data matrix are Gaussian randomvariables whose variances $\eta$ fluctuate from one sample to another accordingto a certain probability density $f(\eta)$ and a single deformation parameter$\gamma$. Three superstatistical classes for $f(\eta)$ are usually considered:$\chi^2$-, inverse $\chi^2$- and log-normal-distributions. While the firstclass, already considered by two of the authors, leads to a power-law decay ofthe spectral density, we here introduce and solve exactly a superposition ofWishart-Laguerre ensembles with inverse $\chi^2$-distribution. Thecorresponding macroscopic spectral density is given by a $\gamma$-deformationof the semi-circle and Mar\v{c}enko-Pastur laws, on a non-compact support withexponential tails. After discussing in detail the validity of Wigner's surmisein the Wishart-Laguerre class, we introduce a generalised $\gamma$-dependentsurmise with stretched-exponential tails, which well approximates theindividual level spacing distribution in the bulk. The analytical results arein excellent agreement with numerical simulations. To illustrate our findingswe compare the $\chi^2$- and inverse $\chi^2$-class to empirical data fromfinancial covariance matrices.
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Abul-Magd AY, Akemann G, Vivo P. Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices. Journal of Physics, A: Mathematical and Theoretical. 2008;42(17).
Abul-Magd, A. Y., Akemann, G., & Vivo, P. (2008). Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices. Journal of Physics, A: Mathematical and Theoretical, 42(17).
Abul-Magd, A. Y., Akemann, G., and Vivo, P. (2008). Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices. Journal of Physics, A: Mathematical and Theoretical 42.
Abul-Magd, A.Y., Akemann, G., & Vivo, P., 2008. Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices. Journal of Physics, A: Mathematical and Theoretical, 42(17).
A.Y. Abul-Magd, G. Akemann, and P. Vivo, “Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices”, Journal of Physics, A: Mathematical and Theoretical, vol. 42, 2008.
Abul-Magd, A.Y., Akemann, G., Vivo, P.: Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices. Journal of Physics, A: Mathematical and Theoretical. 42, (2008).
Abul-Magd, A. Y., Akemann, Gernot, and Vivo, P. “Superstatistical generalizations of Wishart-Laguerre ensembles of random matrices”. Journal of Physics, A: Mathematical and Theoretical 42.17 (2008).
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