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 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 $n^{-\gamma }\log P_{\omega }(X_n<xn)$ 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 $n^{-1+1/s}\log P_{\omega }(X_n<xn)$ oscillates between $0$ and $-\infty$, almost surely. This had previously been shown by Gantert only in a very special case of random environments.

24 pages, 1 figure