An important excited random walk counter-example

TL;DR

This paper constructs a specific excited random walk example in an ergodic environment with infinite cookies per site, demonstrating limitations of previous positive speed conditions in such environments.

Contribution

It provides a counter-example showing that a known positive speed condition does not extend to ergodic random environments.

Findings

Constructed an excited random walk with zero speed in ergodic environment

Confirmed the non-extension of previous positive speed conditions

Demonstrated infinite cookies per site in the example

Abstract

In this paper, we give a detailed construction of an example of excited random walk with speed zero in an ergodic random environment that have an infinite average number of cookies in each site. This example confirms that a result of Mountford, Pimentel and Valle (2006), which gives a sufficient condition for excited random walks in deterministic environment to have positive speed, can not be extended to ergodic random environment.