For fixed m ≥ 1, we consider the product of m independent n × n random matrices with iid entries as n → ∞. Under suitable assumptions on the entries of each matrix, it is known that the limiting empirical distribution of the eigenvalues is described by the m-th power of the circular law. Moreover, this same limiting distribution continues to hold if each iid random matrix is additively perturbed by a bounded rank deterministic error. However, the bounded rank perturbations may create one or more outlier eigenvalues. We describe the asymptotic location of the outlier eigenvalues, which extends a result of Tao [60] for the case of a single iid matrix. Our methods also allow us to consider several other types of perturbations, including multiplicative perturbations. 42 12. Proofs of results from Section 3 54 Appendix

No abstract

Consider the product X = X 1 · · · Xm of m independent n × n iid random matrices. When m is fixed and the dimension n tends to infinity, we prove Gaussian limits for the centered linear spectral statistics of X for analytic test functions. We show that the limiting variance is universal in the sense that it does not depend on m (the number of factor matrices) or on the distribution of the entries of the matrices. The main result generalizes and improves upon previous limit statements for the linear spectral statistics of a single iid matrix by Rider and Silverstein as well as Renfrew and the second author.

No abstract