Empirical Distributions of Eigenvalues of Product Ensembles

Abstract: Assume a finite set of complex random variables form a determinantal point process, we obtain a theorem on the limit of the empirical distribution of these random variables. The result is applied to %We study the limits of the empirical distributions of the eigenvalues of two types of $n$ by $n$ random matrices as $n$ goes to infinity. The first one is the product of $m$ i.i.d. (complex) Ginibre ensembles, and the second one is the product of truncations of $m$ independent Haar unitary matrices with sizes $n_j\times n_j$ for $1\leq j \leq m$. Assuming $m$ depends on $n$, by using the special structures of the eigenvalues we developed, explicit limits of spectral distributions are obtained regardless of the speed of $m$ compared to $n$. For the product of $m$ Ginibre ensembles, as $m$ is fixed, the limiting distribution is known by various authors, e.g., G\"{o}tze and Tikhomirov (2010), Bordenave (2011), O'Rourke and Soshnikov (2011) and O'Rourke {\it et al}. (2014). Our results hold for any $m\geq 1$ which may depend on $n$. For the product of truncations of Haar-invariant unitary matrices, we show a rich feature of the limiting distribution as $n_j/n$'s vary. In addition, some general results on arbitrary rotation-invariant determinantal point processes are also derived. Especially, we obtain an inequality for the fourth moment of linear statistics of complex random variables forming a determinantal point process. This inequality is known for the complex Ginibre ensemble only [Hwang (1986)]. Our method is the determinantal point process rather than the contour integral by Hwang.