  Low-lying states in nuclei are investigated using an ensemble of random
interactions. Both in the nuclear shell model and in the interacting boson
model we find a dominance of $J^P=0^+$ ground states. It is shown that this
feature is not due to time reversal symmetry. In the shell model, evidence is
found for the occurrence of pairing properties, and in the interacting boson
model for both vibrational and rotational band structures. Our results suggest
that these features represent general and robust properties of the model space,
and do not depend on details of the interactions.
