  This work addresses potential theoretic questions for the standard nearest
neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large class of
subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic
measure of $A$, the mean hitting time of $A$, and the Laplace transform of this
hitting time. In particular, we give precise sufficient conditions for the
harmonic measure to be asymptotically uniform, and for the hitting time to be
asymptotically exponentially distributed, as $N\to\infty$. Our approach relies
on a $d$-dimensional extension of the Ehrenfest urn scheme called lumping and
covers the case where $d$ is allowed to diverge with $N$ as long as
$d\leq\alpha_0\frac{N}{\log N}$ for some constant $0<\alpha_0<1$.
