Low-lying states in nuclei are investigated using an ensemble of random int
eractions. Both in the nuclear shell model and in the interacting boson mod
el 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 b
oson 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.