Products of independent elliptic random matrices

Abstract

For fixed $m>1$, we study the product of $m$ independent $N\times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-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 and O'Rourke-Soshnikov concerning the product of independent iid random matrices.

31 pages, 2 figures; minor corrections, added a reference

References in corpus (4)

• On the Asymptotic Spectrum of Products of Independent Random Matrices
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• On the spectrum of sum and product of non-hermitian random matrices

Cited by in corpus (3)

• Eigenvector statistics of the product of Ginibre matrices
• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble