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 logarithms of the singular values of the random product of $n$ of these matrices are all of order at least $\epsilon^2n$, both in expectation; and almost surely for large $n$. To prove this, we develop recent work of Gorodetski and Kleptsyn \cite{gorodetski2023nonstationary}. That paper gives a very flexible method, based on relative entropy, for showing that a non-stationary product of matrices in SL(d,R) has a strictly positive Lyapunov exponent. We extend their work in two ways, firstly by making the estimates quantitative in the context of absolutely continuous distributions, giving the universal estimates described above; and secondly by developing a fibered version of their methods, working on flag bundles instead of the projective space to estimate gaps between arbitrary consecutive exponents. Our methods retain much of the flexibility of those of Gorodetski and Kleptsyn, and we hope that they will find application in other related problems.