Asymptotic laws for a class of quasi-periodic Schr\"odinger cocycles at the lowest energy of the spectrum

Let $(\omega, A_E)$ be a quasi-periodic Schr\"odinger cocycle, where $\omega$ is a Diophantine irrational. The potential is assumed to be $C^2$ with a unique non-degenerate minimum, and the coupling constant is assumed to be large. We show that, as the energy approaches the lowest energy of the spectrum from below, the distance between the Oseledets-directions, in projective coordinates, is asymptotically linear. Moreover, we show that the $C^2$-norm of the Oseledets-directions, in projective coordinates, grows asymptotically (almost) like the inverse of the square root of the distance. Both of these results confirm numerical observations.