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We study the connectivity of random subgraphs of the $d$-dimensional Hamminggraph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bondpercolation on $H(d,n)$ with parameter $p$. We identify the window of thetransition: when $ np- \log n \to - \infty$ the probability that the graph isconnected 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 phasetransition of the giant connected component the Hamming graph (see [Bor et al,2005]). Within the critical window, the connectivity probability does depend ond. We determine how.

Connectivity Threshold for random subgraphs of the Hamming graph