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

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 $λ$ we have $\limsup_{n\rightarrow\infty}\frac{X_n}{n}\leq (2p-1)λ$, 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.

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