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

TL;DR

This paper studies the precise asymptotic behavior of slowdown probabilities in one-dimensional random walks in random environments, revealing oscillations in the decay rate that were previously only partially understood.

Contribution

It demonstrates that the logarithm of slowdown probabilities oscillates between zero and negative infinity at a specific scale, generalizing prior special-case results.

Findings

Logarithm of slowdown probabilities oscillates between 0 and -infinity.

Oscillations occur at the scale of n^{-1+1/s}.

Results hold almost surely for general environments.

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 $n^{-\gamma} \log P_\omega( X_n < x n)$ 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 by Gantert only in a very special case of random environments.