Superstatistical generalisations of Wishart-Laguerre ensembles of random matrices

Using Beck and Cohen’s superstatistics, we introduce in a systematic way a family of generalised 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: χ-, inverse χand log-normal-distribution. While the first class, already considered by two of the authors, leads to a powerlaw decay, we here introduce and solve exactly a superposition of Wishart-Laguerre ensembles with inverse χdistribution. 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. Using a Wigner surmise, we also compute the microscopic level spacing distribution, which displays a stretched exponential tail.