Spectrum of Non-Hermitian Heavy Tailed Random Matrices

Abstract

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 $${\mathbb{C}}$$ depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix $${(a_n^{-1}X_{jk})_{1\leq j,k\leq n}}$$ 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.