Empirical Distributions of Eigenvalues of Product Ensembles

Tiefeng Jiang, Yongcheng Qi

Published 2015-08-13Version 1

We study the limits of the empirical distributions of the eigenvalues of two $n$ by $n$ 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 that 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 of the two matrices we developed, explicit limits of spectral distributions are obtained regardless of the speed of $m$ compared to $n$. In particular, we show a rich feature of the limits for the second matrix as $n_j/n$'s vary. Some general results on arbitrary rotation-invariant determinantal point processes are also derived.