Abstract
We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and perturbative, and the corresponding RWRE has non trivial probability to wander off to the upper right. This is in contrast to the 0-1-law that holds for i.i.d. environments.
Highlights
We construct a two-dimensional counterexample of a random walk in random environment (RWRE)
One might ask for a random environment to satisfy, with 0 ≤ κ < 1/2 some ellipticity constant, the condition
We prove the streetgrid to be stationary and mixing in the Subsections 3.3 and 3.4. These properties are inherited in the definition of the random environment
Summary
We start by fixing the notation and the basic notions of the model. We work in the d-dimensional space Zd, d ≥ 1. As for d ≥ 3, Bramson, Zeitouni and Zerner [1] have a uniformly elliptic, stationary, totally ergodic, and even mixing example of an environment satisfying (1.5). For any ε > 0, there is an ε–perturbative, stationary, mixing random environment ω = Z (ωu)u∈ 2 with associated probability measure P such that for the associated random walk ((Xt), P0ω), it holds that. We prove the streetgrid to be stationary and mixing in the Subsections 3.3 and 3.4 These properties are inherited in the definition of the random environment. In Subsection 3.2, we show that there are areas growing in the direction of 1 that are in some sense large This has the consequence, via the placement of the transition probabilities, that the random walk has positive probability of never leaving these areas, while wandering off to infinity in the direction of 1. By using Poisson processes of different intensities as the underlying structure instead of their“windows” of fixed length, we were able to avoid some of the rigidity of their model and to make assertions on mixing, at the price of developing a completely new construction
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