Superstatistical generalisations of Wishart-Laguerre ensembles of random matrices

Abstract

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: $χ^2$-, inverse $χ^2$- 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 $χ^2$-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 generalised $γ$-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 $χ^2$- and inverse $χ^2$-class to empirical data from financial covariance matrices.