The component structure of dense random subgraphs of the hypercube

Given $p\in (0,1)$, we let $Q_p=Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d\to \mathrm{\infty }$, if $p>\frac{1}{2}$ then with high probability $Q_p$ is connected; and if $p<\frac{1}{2}$ then with high probability $Q_p$ consists of one giant component together with many smaller components which form the `fragment’. Here we fix $p \in (0,\frac12)$, and investigate the fragment, and how it sits inside the hypercube. In particular we give asymptotic estimates for the mean numbers of components in the fragment of each size, and describe their asymptotic distributions and indeed their joint distribution, much extending earlier work of Weber.