Electronic Communications in Probability

The zero-one law for planar random walks in i.i.d. random environments revisited

Martin Zerner

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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.

Article information

Source
Electron. Commun. Probab., Volume 12 (2007), paper no. 32, 326-335.

Dates
Accepted: 5 October 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465224975

Digital Object Identifier
doi:10.1214/ECP.v12-1314

Mathematical Reviews number (MathSciNet)
MR2342711

Zentralblatt MATH identifier
1128.60090

Subjects
Primary: 60K37: Processes in random environments
Secondary: 60F20: Zero-one laws

Keywords
Random environment random walk RWRE transience zero-one law

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Zerner, Martin. The zero-one law for planar random walks in i.i.d. random environments revisited. Electron. Commun. Probab. 12 (2007), paper no. 32, 326--335. doi:10.1214/ECP.v12-1314. https://projecteuclid.org/euclid.ecp/1465224975


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References

  • M. Bramson, O. Zeitouni and M.P.W. Zerner. Shortest spanning trees and a counterexample for random walks in random environments. Ann. Probab. 34 (2006), no. 3, 821–856.
  • P.A. Ferrari, C. Landim and H. Thorisson. Poisson trees, succession lines and coalescing random walks. Ann. I.H.P. Probab. Stat. 40 (2004), no. 2, 141–152.
  • S.A. Kalikow. Generalized random walk in a random environment. Ann. Probab. 9 (1981), no. 5, 753–768.
  • A.-S. Sznitman and M. Zerner. A law of large numbers for random walks in random environment. Ann. Probab. 27 (1999), no. 4, 1851–1869.
  • O. Zeitouni. Random walks in random environment. Lectures on probability theory and statistics. Lecture Notes in Math. 1837 Springer, Berlin (2004), 189–312.
  • M.P.W. Zerner and F. Merkl. A zero-one law for planar random walks in random environment. Ann. Probab. 29 (2001), no. 4 1716–1732.