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

Hua-Ming Wang

Published 2016-02-09Version 1

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.