Abstract
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.
Highlights
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 Z2
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
Let us first recall the model of random walks in random environments (RWRE), see [Zei04] for a survey
Summary
Let us first recall the model of random walks in random environments (RWRE), see [Zei04] for a survey. By adjusting zL and using d = 2 one can force the paths of the two walkers to intersect at some point x in the middle slab with a positive probability, which is bounded away from 0 uniformly in L In this step some technical result [ZerMe01, Lemma 7] about sums of four independent random variables is used. This way the technical lemma [ZerMe01, Lemma 7] is not needed anymore and general directions l ∈/ {e1, e2} can be handled more .
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