Lyapunov exponents of orthogonal-plus-normal cocycles

TL;DR

This paper derives exact formulas and asymptotic behavior for Lyapunov exponents of matrix products combining orthogonal matrices with small Gaussian perturbations, revealing their dependence on the perturbation size.

Contribution

It provides a novel exact expression for Lyapunov exponents of orthogonal-plus-normal cocycles and analyzes their asymptotic behavior as the perturbation parameter approaches zero.

Findings

Exact formula for the jth Lyapunov exponent in terms of Gram-Schmidt orthogonalization.

Asymptotic expansion of Lyapunov exponents showing quadratic dependence on epsilon.

Demonstration of the exponents' behavior as epsilon tends to zero.

Abstract

We consider products of matrices of the form $A_n=O_n+\epsilon N_n$ where $O_n$ is a sequence of $d\times d$ orthogonal matrices and $N_n$ has independent standard normal entries and the $(N_n)$ are mutually independent. We study the Lyapunov exponents of the cocycle as a function of $\epsilon$, giving an exact expression for the $j$th Lyapunov exponent in terms of the Gram-Schmidt orthogonalization of $I+\epsilon N$. Further, we study the asymptotics of these exponents, showing that $\lambda_j=(d-2j)\epsilon^2/2+O(\epsilon^4|\log\epsilon|^4)$.