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 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 $1\le j \le m$. 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., Götze and Tikhomirov (On the asymptotic spectrum of products of independent random matrices, 2010. http://arxiv.org/pdf/1012.2710v3.pdf), Bordenave (Electron Commun Probab 16:104–113, 2011), O’Rourke and Soshnikov (Electron J Probab 16(81):2219–2245, 2011) and O’Rourke et al. (J Stat Phys 160(1):89–119, 2015). Our results hold for any $m\ge 1$ 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. In particular, 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 in Random matrices and their applications (Brunswick, Maine, 1984), Contemporary Mathematics, American Mathematics Society, Providence, vol 50, pp 145–152, 1986). Our method is the determinantal point process rather than the contour integral by Hwang.