Zero-one law for directional transience of one dimensional excited random walks

TL;DR

This paper proves a zero-one law for the directional transience of one-dimensional excited random walks in stationary ergodic environments, resolving an open problem in the field.

Contribution

It establishes a zero-one law for the probability of transience direction in excited random walks, a problem previously posed by Kosygina and Zerner.

Findings

The probability of transience to the right or left is almost surely zero or one.

The result applies to stationary ergodic and elliptic cookie environments.

It resolves a longstanding open problem in the theory of excited random walks.

Abstract

The probability that a one dimensional excited random walk in stationary ergodic and elliptic cookie environment is transient to the right (left) is either zero or one. This solves a problem posed by Kosygina and Zerner [8].