Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed ${\lim }_{n\to \mathrm{\infty }}\frac{X_n}{n}=v_{\alpha }>0$. Gantert and Zeitouni showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_{\omega }(X_n<xn)$ with $x \in (0,v_\alpha)$decayapproximatelylike$\exp{-n^{1-1/s}}$ for a deterministic $s>1$. More precisely, they showed that $n^{-\gamma} \log P_\omega( X_n < x n)$convergesto$0$or$-\infty$dependingonwhether$\gamma > 1-1/s$or$\gamma < 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1\mathrm{/}s}\log P_{\omega }(X_n<xn)$ oscillates between $0$ and $-\mathrm{\infty }$, almost surely. This had previously been shown by Gantert only in a very special case of random environments.