On Maximizing the Speed of a Random Walk in Fixed Environments

TL;DR

This paper investigates how to optimize the placement of p-drifts in a fixed environment to minimize the expected hitting time of a random walk, revealing that equally spacing these drifts asymptotically achieves the fastest hitting time.

Contribution

It provides a theoretical analysis of the optimal configuration of p-drifts in a fixed environment to maximize the speed of a random walk.

Findings

Equally spaced p-drifts asymptotically minimize hitting time.

The expected hitting time depends on the arrangement of p-drifts.

Optimal spacing improves the efficiency of the random walk.

Abstract

We consider a random walk in a fixed Z environment composed of two point types: (q,1-q) and (p,1-p) for 1/2<q<p. We study the expected hitting time at N for a given number k of p-drifts in the interval [1,N-1], and find that this time is minimized asymptotically by equally spaced p-drifts.