The need for speed : Maximizing random walks speed on fixed environments

TL;DR

This paper investigates the maximum possible speed of nearest neighbor random walks on fixed one-dimensional environments with two types of points, establishing bounds and identifying optimal configurations for speed.

Contribution

It provides a bound on the asymptotic speed of the walk based on environment density and identifies the environment configuration that maximizes speed.

Findings

The asymptotic speed is bounded by (2p-1)λ.

Equally spaced drifts optimize the walk's speed.

The environment with evenly spaced drifts achieves near-optimal speed.

Abstract

We study nearest neighbor random walks on fixed environments of $\mathbb{Z}$ composed of two point types : $(1/2,1/2)$ and $(p,1-p)$ for $p>1/2$. We show that for every environment with density of $p$ drifts bounded by $\lambda$ we have $\limsup_{n\rightarrow\infty}\frac{X_n}{n}\leq (2p-1)\lambda$, where $X_n$ is a random walk on the environment. In addition up to some integer effect the environment which gives the best speed is given by equally spaced drifts.