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.

arXiv comment31 pages, 2 figures; minor corrections, added a reference

References in corpus (19)

• Products of Rectangular Random Matrices: Singular Values and Progressive Scattering
• Around the circular law
• Spectrum of the Product of Independent Random Gaussian Matrices
• The circular law for random matrices
• Eigenvalues and Singular Values of Products of Rectangular Gaussian Random Matrices
• Singular value correlation functions for products of Wishart random matrices
• Convergence of the largest singular value of a polynomial in independent Wigner matrices
• $R$-diagonal pairs - a common approach to Haar unitaries and circular elements
• Universal microscopic correlation functions for products of independent Ginibre matrices
• On the Asymptotic Spectrum of Products of Independent Random Matrices
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• A new application of Random Matrices: Ext(C*_{red}(F_2)) is not a group
• Hole probabilities and overcrowding estimates for products of complex Gaussian matrices
• Elliptic law for real random matrices
• On one generalization of the elliptic law for random matrices
• Low rank perturbations of large elliptic random matrices
• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• On the spectrum of sum and product of non-hermitian random matrices
• On the concentration of random multilinear forms and the universality of random block matrices

Cited by in corpus (20)

• Recent exact and asymptotic results for products of independent random matrices
• Products of Independent Gaussian Random Matrices
• Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method
• Eigenvalues of block structured asymmetric random matrices
• On fluctuations of eigenvalues of random band matrices
• A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
• Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application
• Eigenvector statistics of the product of Ginibre matrices
• On one generalization of the elliptic law for random matrices
• Permanental processes from products of complex and quaternionic induced Ginibre ensembles
• Time-inhomogeneous random Markov chains
• Empirical Distributions of Eigenvalues of Product Ensembles
• Universality for products of random matrices I: Ginibre and truncated unitary cases
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth
• Limiting Empirical Spectral Distribution for Products of Rectangular Matrices
• Low-degree factors of random polynomials
• Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble
• Spectral Radii of Products of Random Rectangular Matrices
• Finite rank perturbation of non-Hermitian random matrices: heavy tail and sparse regimes
• Product of Non-Hermitian Random Matrices