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

TL;DR

This paper simplifies the proof of the zero-one law for directional transience in planar i.i.d. random walks and constructs a counterexample with improved ergodic properties.

Contribution

It provides a simplified proof of an existing zero-one law and introduces a new counterexample with better ergodic properties in non-uniform environments.

Findings

Simplified proof of the zero-one law for planar RWRE

Construction of a counterexample with improved ergodic properties

Illustration of the law's applicability in non-uniform environments

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.