On maximizing the speed of a random walk in fixed environments

Abstract

We consider a random walk in a fixed $\mathbb{Z}$ environment composed of two point types: $q$-drifts (in which the probabiliy to move to the right is $q$, and $1-q$ to the left) and $p$-drifts, where $\frac{1}{2}<q<p$. We study the expected hitting time of a random walk at $N$ given the number of $p$-drifts in the interval $[1,N-1]$, and find that this time is minimized asymptotically by equally spaced $p$-drifts.