A Stationary, mixing and perturbative counterexample to the 0-1-law for random walk in random environment in two dimensions

Hadrian Heil
2013 Electronic Journal of Probability  
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. Abstract We construct a two-dimensional counterexample of a random walk in random environment (RWRE). The environment is stationary, mixing and ε-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. Here, 1 denotes the vector (1, 1). A stationary, mixing counterexample A preprint by Guo [2] is concerned
more » ... 2] is concerned with the limiting velocity of the random walk in random environment on the events in the case where the random environment satisfies uniform ellipticity and a certain strong mixing condition which holds in Gibbsian environments, for instance. Proof of Theorem 1.1, and organisation of the article. In Section 2, we construct an object called streetgrid which we use to define the actual random environment in Subsection 2.3. 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. This is shown in Subsection 4. The same arguments could be repeated for − 1, which finishes the proof. We should want to indicate some of the sources of inspiration that contributed to this article. The ideas of conducting the random walk to infinity on a "treelike structure" of "not too slowly growing roads leading to infinity" has been applied in [1] . As for how to construct such a structure in dimension d = 2, Häggström and Mester [3] had the idea of ever larger, ever rarer streets joining each other. 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.
doi:10.1214/ejp.v18-1880 fatcat:bydexn5zlra7vaha2jfdnw3ml4