Gaussian fluctuations for products of random matrices

Abstract

We study global fluctuations for singular values of M-fold products of several right-unitarily invariant N × N random matrix ensembles. As N → ∞, we show the fluctuations of their height functions converge to an explicit Gaussian field, which is log-correlated for M fixed and has a white noise component for M → ∞ jointly with N. Our technique centers on the study of the multivariate Bessel generating functions of these spectral measures, for which we prove a central limit theorem for global fluctuations via certain conditions on the generating functions. We apply our approach to a number of ensembles, including square roots of Wishart, Jacobi, and unitarily invariant positive definite matrices with fixed spectrum, using a detailed asymptotic analysis of multivariate Bessel functions to verify the necessary conditions.