Spectrum of Non-Hermitian Heavy Tailed Random Matrices

Let (Xjk)j,k ≥ 1 be i.i.d. complex random variables such that |Xjk| 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 Xjk, we prove that there exist a deterministic sequence an ~ n1/α and a probability measure μα on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document} depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(a_n^{-1}X_{jk})_{1\leq j,k\leq n}}$$\end{document} converges weakly to μα 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 μα. In contrast with the Hermitian case, we find that μα is not heavy tailed.