Spectrum of non-Hermitian heavy tailed random matrices

Understand

Let (X_{jk})_{j,k>=1} be i.i.d.

• complex random variables such that |X_{jk}| is in the domain of attraction of an alpha-stable law, with 0< alpha <2.
• Our main result is a heavy tailed counterpart of Girko's circular law.
• Namely, under some additional smoothness assumptions on the law of X_{jk}, we prove that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability measure mu_alpha on C depending only on alpha such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1} (X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity.