A zero-one law for planar random walks in random environment
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.
scientific article; zbMATH DE number 1906037