On the slowdown of random walk in random environment with bounded jumps

TL;DR

This paper demonstrates that under certain conditions, a transient random walk with bounded jumps in a random environment grows significantly slower than linear speed, with the position scaled by a power less than one converging to zero.

Contribution

It establishes a new slow-down phenomenon for transient random walks with bounded jumps in random environments, showing sublinear growth rates.

Findings

Random walk growth is slower than linear under certain assumptions.

Almost sure convergence of scaled position to zero for powers less than one.

Existence of a parameter 0<s<1 characterizing the slowdown.

Abstract

In this paper we prove that under certain assumptions the transient random walk in random environment with bounded jumps (in $\mathbb{Z}$) grows much slower than the speed $n$. Precisely, there is $0<s<1$, such that although $X_n\rto$ we have $\frac{X_n}{n^{s'}}\rightarrow 0$ for $0<s<s'$ almost surely.