A Zero-One Law for Planar Random walks in Random Environment

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Quenched invariance principle for simple random walk on percolation clusters (2005, methodology): “The first counterexample, based on constructions in [43] and [11], was provided to us by Martin Zerner.”
A law of large numbers for random walks in random mixing environments (2002, background): “A motivation for this question relates to an example in [22], Proposition 2, of a RWRE in dimension 2 with the nonstandard asymptotics: P(limn Xn/n = w) = 1/2, P(limn Xn/n = −w) = 1/2 for some nonzero vector w.”
Random Walks in Random Environment (2009, background): “Our proof borrows from [82], which, still in the i.”
Large deviations for random walks in a random environment (2003, background): “A zero-one law of Zerner and Merkl [9] proves in d = 2 (d = 1 is trivial) that for any w ∈ W, in particular for w = φ, U (w) = 0 or 1.”