Lyapunov exponents of orthogonal-plus-normal cocycles

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 ${\lambda }_j=(d-2j){\epsilon }^2\mathrm{/}2+O({\epsilon }^4\mathrm{\mid }\log \epsilon {\mathrm{\mid }}^4)$.