In this note we present a simplified proof of the zero-one law by Merkl and Zerner (2001) for directional transience of random walks in i.i.d. random environments (RWRE) on the square lattice. Also, we indicate how to construct a two-dimensional counterexample in a non-uniformly elliptic and stationary environment which has better ergodic properties than the example given by Merkl and Zerner.
A zero-one law for random walks in random environments on $\mathbb{Z}^2$ with bounded jumps
Random walks in random environments without ellipticity
Shortest spanning trees and a counterexample for random walks in random environments
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