Spectrum of Non-Hermitian Heavy Tailed Random Matrices

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 α-stable law, with 0 < α < 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 ~ n ^1/α and a probability measure μ _α on \mathbbC{\mathbb{C}} depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix (a_n^-1X_jk)_{1 ￡ j,k ￡ n}{(a_n^{-1}X_{jk})_{1\leq j,k\leq n}} converges weakly to μ _α as n → ∞. 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 μ _α . In contrast with the Hermitian case, we find that μ _α is not heavy tailed.