  We compare eigenvalue densities of Wigner random matrices whose elements are
independent identically distributed (iid) random numbers with a Levy
distribution and maximally random matrices with a rotationally invariant
measure exhibiting a power law spectrum given by stable laws of free random
variables. We compute the eigenvalue density of Wigner-Levy (WL) matrices using
(and correcting) the method by Bouchaud and Cizeau (BC), and of free random
Levy (FRL) rotationally invariant matrices by adapting results of free
probability calculus. We compare the two types of eigenvalue spectra. Both
ensembles are spectrally stable with respect to the matrix addition. The
discussed ensemble of FRL matrices is maximally random in the sense that it
maximizes Shannon's entropy. We find a perfect agreement between the
numerically sampled spectra and the analytical results already for matrices of
dimension N=100. The numerical spectra show very weak dependence on the matrix
size N as can be noticed by comparing spectra for N=400. After a pertinent
rescaling spectra of Wigner-Levy matrices and of symmetric FRL matrices have
the same tail behavior. As we discuss towards the end of the paper the
correlations of large eigenvalues in the two ensembles are however different.
We illustrate the relation between the two types of stability and show that the
addition of many randomly rotated Wigner-Levy matrices leads by a matrix
central limit theorem to FRL spectra, providing an explicit realization of the
maximal randomness principle.
