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

Abstract

We consider a one dimensional random walk in a random environment (RWRE) with a positive speed $\lim _{n\to \infty }\frac{X_n} {n}=v_\alpha >0$. Gantert and Zeitouni [9] showed that if the environment has both positive and negative local drifts then the quenched slowdown probabilities $P_\omega (X_n < xn)$ with $x \in (0,v_\alpha )$ decay approximately like $\exp \{-n^{1-1/s}\}$ for a deterministic $s > 1$. More precisely, they showed that $n^{-\gamma } \log P_\omega ( X_n < x n)$ converges to $0$ or $-\infty $ depending on whether $\gamma > 1-1/s$ or $\gamma < 1-1/s$. In this paper, we improve on this by showing that $n^{-1+1/s} \log P_\omega ( X_n < x n)$ oscillates between $0$ and $-\infty $, almost surely. This had previously been shown only in a very special case of random environments [7].