Spectrum of Non-Hermitian Heavy Tailed Random Matrices

Let (X(jk))(j,k >= 1) be i.i.d complex random variables such that vertical bar X(jk)vertical bar 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 exist a deterministic sequence a(n) similar to 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 -> infinity. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of mu(alpha). In contrast with the Hermitian case, we find that mu(alpha) is not heavy tailed.