  We study the connectivity of random subgraphs of the $d$-dimensional Hamming
graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$
vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond
percolation on $H(d,n)$ with parameter $p$. We identify the window of the
transition: when $ np- \log n \to - \infty$ the probability that the graph is
connected goes to $0$, while when $ np- \log n \to + \infty$ it converges to
$1$.
  We also investigate the connectivity probability inside the critical window,
namely when $ np- \log n \to t \in \mathbb{R}$.
  We find that the threshold does not depend on $d$, unlike the phase
transition of the giant connected component the Hamming graph (see [Bor et al,
2005]). Within the critical window, the connectivity probability does depend on
d. We determine how.
