Unexpected results in asymptotically free quantum field theories

We study the behavior of asymptotically free (AF) spin and gauge models when their continuous symmetry group is replaced by different discrete non-Abelian subgroups. Precise numerical results with relative errors down to O(0.1%) suggest that the models with large subgroups are in the universality class of the underlying original models. We argue that such a scenario is consistent with the known properties of AF theories. The small statistical errors allow a detailed investigation of the cut-off effects also. At least up to correlation lengths ~300 they follow effectively an O(a) rather than the expected O(a^2) form both in the O(3) and in the dodecahedron model.