  We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is
the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the
edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In
Borgs et al., Random subgraphs of finite graphs: I. The scaling window under
the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random
subgraphs of finite graphs: II. The lace expansion and the triangle condition,
Ann. Probab. (2005), the size of the largest connected component was estimated
precisely for a large class of graphs including H(2,n) for $\vep\leq \Lambda
V^{-1/3}$, where $\Lambda > 0$ is a constant and $V=n^2$ denotes the number of
vertices in H(2,n). Until now, no matching lower bound on the size in the
supercritical regime has been obtained.
  In this paper we prove that, when $\vep\gg (\log{V})^{1/3} V^{-1/3}$, then
the largest connected component has size close to $2\vep V$ with high
probability. We thus obtain a law of large numbers for the largest connected
component size, and show that the corresponding values of $p$ are
supercritical. Barring the factor $(\log{\chs{V}})^{1/3}$, this identifies the
size of the largest connected component all the way down to the critical $p$
window.
