This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.
Heavy-tailed random matrices