Superstatistical generalizations of Wishart–Laguerre ensembles of random matrices

Abstract

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.