Range of (1,2) random walk in random environment

TL;DR

This paper studies a (1,2) random walk in a random environment, showing that its range grows linearly but covers only a fraction of the positive integers, unlike nearest neighbor walks.

Contribution

It proves that the range of a (1,2) walk in random environment covers a linear but fractional portion of the positive integers, a phenomenon absent in nearest neighbor walks.

Findings

Range grows linearly with position x

Fraction of covered sites converges to a constant between 0 and 1

Distinct behavior from nearest neighbor random walks

Abstract

Consider $(1,2)$ random walk in random environment $\{X_n\}_{n\ge0}.$ In each step, the walk jumps at most a distance $2$ to the right or a distance $1$ to the left. For the walk transient to the right, it is proved that almost surely $\lim_{x\rightarrow\infty}\frac{\#\{X_n:\ 0\le X_n\le x,\ n\ge0\}}{x}=\theta$ for some $0<\theta<1.$ The result shows that the range of the walk covers only a linear proportion of the lattice of the positive half line. For the nearest neighbor random walk in random or non-random environment, this phenomenon could not appear in any circumstance.