On the Speed of an Excited Asymmetric Random Walk

TL;DR

This paper investigates the speed of excited asymmetric random walks, providing new proofs and explicit formulas for their limiting speed, which depends on initial conditions and a parameter .

Contribution

It introduces a generalized excited asymmetric random walk with bias and derives explicit formulas for its limiting speed under certain conditions.

Findings

Explicit formula for the limiting speed of excited asymmetric random walks.

The speed depends on initial conditions and a parameter .

A new proof relating  and the speed is provided.

Abstract

An excited random walk is a non-Markovian extension of the simple random walk, in which the walk's behavior at time $n$ is impacted by the path it has taken up to time $n$. The properties of an excited random walk are more difficult to investigate than those of a simple random walk. For example, the limiting speed of an excited random walk is either zero or unknown depending on its initial conditions. While its limiting speed is unknown in most cases, the qualitative behavior of an excited random walk is largely determined by a parameter $\delta$ which can be computed explicitly. Despite this, it is known that the limiting speed cannot be written as a function of $\delta$. We offer a new proof of this fact, and use techniques from this proof to further investigate the relationship between $\delta$ and speed. We also generalize the standard excited random walk by introducing a "bias" to the right, and call this generalization an excited asymmetric random walk. Under certain initial conditions we are able to compute an explicit formula for the limiting speed of an excited asymmetric random walk.