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In this note we present a simplified proof of the zero-one law by Merkl andZerner (2001) for directional transience of random walks in i.i.d. randomenvironments (RWRE) on the square lattice. Also, we indicate how to construct atwo-dimensional counterexample in a non-uniformly elliptic and stationaryenvironment which has better ergodic properties than the example given by Merkland Zerner.

The zero-one law for planar random walks in i.i.d. random environments revisited