Lyapunov exponents of orthogonal-plus-normal cocycles

We consider products of matrices of the form An=On+epsilon Nn where (On)n >= 1 is a sequence of d x d orthogonal matrices, Nn has independent standard normal entries and where the (Nn)n >= 1 are mutually independent. We study the Lyapunov exponents of the cocycle as a function of epsilon, giving an exact expression for the jth 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).