A zero-one law for planar random walks in random environment. (English) Zbl 1016.60093

The main purpose of the paper is the derivation of a zero-one law for random walks \((X_n)_n\) in i.i.d. random environment on \(Z^2\). It is shown that for any fixed direction \(l\in R^2\setminus \{0\}\) the event that the inner product \(X_nl\) tends to \(+\infty\) as \(n\to \infty\) has probability either 0 or 1. This is a solution of the problem posed by S. A. Kalikow whether the event that the \(x\)-coordinate of a random walk in a two-dimensional random environment approaches \(\infty\) has necessarily probability either zero or one.


60K37 Processes in random environments
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60F20 Zero-one laws
Full Text: DOI


[1] den Hollander, F. and Thorisson, H. (1994). Shift-coupling and a zero-one law for random walk in random environment. Acta Appl. Math. 34 37-50. · Zbl 0804.60055
[2] Kalikow, S. A. (1981). Generalized random walk in a random environment. Ann. Probab. 9 753- 768. · Zbl 0545.60065
[3] Soucaliuc, F., T óth, B. and Werner, W. (2000). Reflection and coalescence between independent one-dimensional Brownian paths. Ann. Inst. H. Poincaré Probab. Statist. 36 509-545. · Zbl 0968.60072
[4] Sznitman, A. S. and Zerner, M. P. W. (1999). A law of large numbers for random walks in random environment. Ann. Probab. 27 1851-1869. · Zbl 0965.60100
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.