The need for speed: Maximizing the speed of random walk in fixed environments

We study nearest neighbor random walks in fixed environments of 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 λ we have lim sup _n→∞ Xn/n ≤ (2p-1)λ, where X _n is a random walk in the environment. In addition up to some integer effect the environment which gives the greatest speed is given by equally spaced drifts.