Equality of Lyapunov and Stability Exponents for Products of Isotropic Random Matrices

In the study of random matrices, Lyapunov exponents characterize the rate of exponential growth of the singular values of product of increasing number of random matrices of fixed order. These have been well studied objects in the literature. Only recently, absolute values of the eigenvalues of such products of random matrices have come under analytical study. The quantities that characterize the rate of exponential growth of the absolute values of eigenvalues have been named stability exponents, in order to distinguish them from that of the singular values. In the cases of Ginibre matrices and truncated Haar unitary matrices, the stability exponents have been observed to match with the Lyapunov exponents. In this article, we generalize this result to the case of all isotropic random matrices. We also derive the asymptotic joint probability distributions for the fluctuations of both the singular values and the absolute values of eigenvalues of product of increasing number of real or complex isotropic random matrices of fixed order. Moreover, Lyapunov exponents are distinct, unless the random matrices are random scalar multiples of Haar unitary matrices or orthogonal matrices. As a corollary of this, we show the probability that the product of real isotropic random matrices has all the eigenvalues real goes to one as n -> infinity.