Asymptotics of finite system Lyapunov exponents for some random matrix ensembles

Abstract

For products PN of N random matrices of size d x d, there is a natural notion of finite N Lyapunov exponents {μi}i=1d. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for μ1, methods known for the computation of lim N→∞ 〈μi〉 are used to compute the large N form of the variances of the exponents. Analogous calculations are performed in the case that the matrices making up PN are products of sub-blocks of random unitary matrices with Haar measure. Furthermore, we make some remarks relating to the coincidence of the Lyapunov exponents and the stability exponents relating to the eigenvalues of PN.