Upper and Lower Bounds on the Speed of a One Dimensional Excited Random Walk

TL;DR

This paper establishes the first meaningful bounds on the speed of one-dimensional excited random walks, a complex self-interacting process, providing insights into when the walk's speed is non-zero and how close the bounds are.

Contribution

It derives the first explicit non-trivial upper and lower bounds for the speed of excited random walks, advancing understanding of their long-term behavior.

Findings

Bounds are close in certain cases

Speed is non-zero under specific conditions

Provides a framework for analyzing ERW speed

Abstract

Excited random walks (ERWs) are a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk, defined as $V = \lim_{n \rightarrow \infty} \frac{X_n}{n}$ where $X_n$ is the state of the walk at time $n$. While results exist that indicate when the speed is non-zero, there exists no explicit formula for the speed. It is difficult to solve for the speed directly due to complex dependencies in the walk since the next step of the walker depends on how many times the walker has reached the current site. We derive the first non-trivial upper and lower bounds for the speed of the walk. In certain cases these upper and lower bounds are remarkably close together.