Ordered states of adatoms on graphene

We show that a dilute ensemble of epoxy-bonded adatoms on graphene has a tendency to form a spatially correlated state accompanied by a gap in graphene's electron spectrum. This effect emerges from the electron-mediated interaction between adatoms with a peculiar $1/r^3$ distance dependence. The partial ordering transition is described by a random bond three state Potts model.