Effective Gaps between singular values of non-stationary matrix products subject to non-degenerate noise

Abstract: We study the singular values and Lyapunov exponents of non-stationary random matrix products subject to small, absolutely continuous, additive noise. Consider a fixed sequence of matrices of bounded norm. Independently perturb the matrices by additive noise distributed according to Lebesgue measure on matrices with norm less than $\epsilon$. Then the gaps between the log of the singular values of the random product of $n$ of these matrices is of order $\epsilon^2n$ in expectation, and almost surely the gap grows like $\epsilon^2n$.