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.
"The zero-one law for planar random walks in i.i.d. random environments revisited." Electron. Commun. Probab. 12 326 - 335, 2007. https://doi.org/10.1214/ECP.v12-1314