Products of Independent Elliptic Random Matrices

For fixed $$m > 1$$m>1, we study the product of $$m$$m independent $$N \times N$$N×N elliptic random matrices as $$N$$N tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $$1$$1, to the $$m$$m-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves. Our result also generalizes earlier results of Götze–Tikhomirov (On the asymptotic spectrum of products of independent random matrices, available at http://arxiv.org/abs/1012.2710) and O’Rourke–Soshnikov (J Probab 16(81):2219–2245, 2011) concerning the product of independent iid random matrices.