## Electronic Journal of Probability

### 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].

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 16, 27 pp.

Dates
Accepted: 12 February 2016
First available in Project Euclid: 25 February 2016

https://projecteuclid.org/euclid.ejp/1456412956

Digital Object Identifier
doi:10.1214/16-EJP4529

Mathematical Reviews number (MathSciNet)
MR3485358

Zentralblatt MATH identifier
1336.60197

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F10: Large deviations 60J15

#### Citation

Ahn, Sung Won; Peterson, Jonathon. Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment. Electron. J. Probab. 21 (2016), paper no. 16, 27 pp. doi:10.1214/16-EJP4529. https://projecteuclid.org/euclid.ejp/1456412956

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