  A set of localized, non-Abelian anyons - such as vortices in a p_x + i p_y
superconductor or quasiholes in certain quantum Hall states - gives rise to a
macroscopic degeneracy. Such a degeneracy is split in the presence of
interactions between the anyons. Here we show that in two spatial dimensions
this splitting selects a unique collective state as ground state of the
interacting many-body system. This collective state can be a novel gapped
quantum liquid nucleated inside the original parent liquid (of which the anyons
are excitations). This physics is of relevance for any quantum Hall plateau
realizing a non-Abelian quantum Hall state when moving off the center of the
plateau.
