Asymptotics of finite system Lyapunov exponents for some random matrix ensembles

TL;DR

This paper analyzes the asymptotic behavior of finite system Lyapunov exponents for products of random matrices, providing variance calculations for Gaussian and unitary matrix ensembles and discussing their relation to stability exponents.

Contribution

It introduces methods to compute the large N asymptotics of the variances of Lyapunov exponents for specific random matrix ensembles, extending existing techniques.

Findings

Derived large N variance formulas for Gaussian matrices

Extended analysis to products of sub-blocks of Haar-random unitary matrices

Discussed the relationship between Lyapunov and stability exponents

Abstract

For products $P_N$ of $N$ random matrices of size $d \times d$, there is a natural notion of finite $N$ Lyapunov exponents $\{\mu_i\}_{i=1}^d$. In the case of standard Gaussian random matrices with real, complex or real quaternion elements, and extended to the general variance case for $\mu_1$, methods known for the computation of $\lim_{N \to \infty} \langle \mu_i \rangle$ 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 $P_N$ 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 $P_N$.