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

**Hadrian Heil**

*(Technische Universität München)*

#### 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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-33

Publication Date: January 4, 2013

DOI: 10.1214/EJP.v18-1880

#### References

- Bramson, Maury; Zeitouni, Ofer; Zerner, Martin P. W. Shortest spanning trees and a counterexample for random walks in random
environments.
*Ann. Probab.*34 (2006), no. 3, 821--856. MR2243869 - Xiaoqin Guo. On the limiting velocity of random walks in mixing random environment. Preprint.
- Häggström, Olle; Mester, Péter. Some two-dimensional finite energy percolation processes.
*Electron. Commun. Probab.*14 (2009), 42--54. MR2481665 - Mark Holmes and Thomas~S. Salisbury. Degenerate Random Walks in Random Environment. Preprint.
- Kalikow, Steven A. Generalized random walk in a random environment.
*Ann. Probab.*9 (1981), no. 5, 753--768. MR0628871 - Sznitman, Alain-Sol; Zerner, Martin. A law of large numbers for random walks in random environment.
*Ann. Probab.*27 (1999), no. 4, 1851--1869. MR1742891 - Zerner, Martin P. W. The zero-one law for planar random walks in i.i.d. random environments
revisited.
*Electron. Comm. Probab.*12 (2007), 326--335 (electronic). MR2342711 - Zerner, Martin P. W.; Merkl, Franz. A zero-one law for planar random walks in random environment.
*Ann. Probab.*29 (2001), no. 4, 1716--1732. MR1880239

This work is licensed under a Creative Commons Attribution 3.0 License.