  We study heavy-tailed Hermitian random matrices that are unitarily invariant.
The invariance implies that the eigenvalue and eigenvector statistics are
decoupled. The motivating question has been whether a freely stable random
matrix has stable eigenvalue statistics for the largest eigenvalues in the
tail. We investigate this question through the use of both numerical and
analytical means, the latter of which makes use of the supersymmetry method. A
surprising behaviour is uncovered in that a freely stable random matrix does
not necessarily yield stable statistics and if it does then it might exhibit
Poisson or Poisson-like statistics. The Poisson statistics have been already
observed for heavy-tailed Wigner matrices. We conclude with two conjectures on
this peculiar behaviour.
