Quenched Limits for Transient, Ballistic, Sub-Gaussian One-Dimensional Random Walk in Random Environment

TL;DR

This paper investigates the behavior of one-dimensional transient random walks in random environments, revealing that under the quenched law, no universal limit law exists, with specific subsequences exhibiting different limiting distributions.

Contribution

It demonstrates that, unlike the annealed case, the quenched law does not admit a universal limit law, showing the existence of subsequences with different limiting behaviors.

Findings

Quenched law does not have a universal limit law.

Existence of subsequences with different limiting distributions.

Quenched CLT holds along certain subsequences.

Abstract

We consider a nearest-neighbor, one-dimensional random walk $\{X_n\}_{n\geq 0}$ in a random i.i.d. environment, in the regime where the walk is transient with speed v_P > 0 and there exists an $s\in(1,2)$ such that the annealed law of $n^{-1/s} (X_n - n v_P)$ converges to a stable law of parameter s. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible. In particular we show that there exist sequences {t_k} and {t_k'} depending on the environment only, such that a quenched central limit theorem holds along the subsequence t_k, but the quenched limiting distribution along the subsequence t_k' is a centered reverse exponential distribution. This complements the results of a recent paper of Peterson and Zeitouni (arXiv:0704.1778v1 [math.PR]) which handled the case when the parameter $s\in(0,1)$.