Superstatistical generalizations 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 beta = 1, 2 and 4. The entries of the data matrix are Gaussian random variables whose variances eta fluctuate from one sample to another according to 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 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 chi(2)-distribution. The corresponding macroscopic spectral density is given by a gamma-deformation of the semi-circle and Marcenko-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 gamma-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 chi(2)- and inverse chi(2)-classes to empirical data from financial covariance matrices.