  We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor
edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for
which the expected cluster size of a fixed vertex attains the value $\lambda
2^{n/3}$, where $\lambda$ is a small positive constant. Let
$\epsilon=n(p-p_c(n))$. In two previous papers, we showed that the largest
cluster inside a scaling window given by $|\epsilon|=\Theta(2^{-n/3})$ is of
size $\Theta(2^{2n/3})$, below this scaling window it is at most $2(\log2)
n\epsilon^{-2}$, and above this scaling window it is at most $O(\epsilon 2^n)$.
In this paper, we prove that for $p - p_c(n) \geq e^{-cn^{1/3}}$ the size of
the largest cluster is at least $\Theta(\epsilon 2^n)$, which is of the same
order as the upper bound. This provides an understanding of the phase
transition that goes far beyond that obtained by previous authors. The proof is
based on a method that has come to be known as ``sprinkling,'' and relies
heavily on the specific geometry of the $n$-cube.
