  We use first-principles Quantum Monte-Carlo simulations to study the
Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between hydrogen adatoms
attached to a graphene sheet. We find that the pairwise RKKY interactions at
distances of a few lattice spacings are strongly affected by inter-electron
interactions, in particular, the potential barrier between widely separated
adatoms and the dimer configuration becomes wider and thus harder to penetrate.
We also point out that anti-ferrromagnetic and charge density wave orderings
have very different effects on the RKKY interaction. Finally, we analyze the
stability of several regular adatom superlattices with respect to small
displacements of a single adatom, distinguishing the cases of adatoms which
populate either both or only one sublattice of the graphene lattice.
