Discrete Symmetry Enhancement in Nonabelian Models and the Existence of Asymptotic Freedom

We study the universality between a discrete spin model with icosahedral symmetry and the O(3) model in two dimensions. For this purpose we study numerically the renormalized two-point functions of the spin field and the four point coupling constant. We find that those quantities seem to have the same continuum limits in the two models. This has far reaching consequences, because the icosahedron model is not asymptotically free in the sense that the coupling constant proposed by L"uscher, Weisz and Wolff [1] does not approach zero in the short distance limit. By universality this then also applies to the O(3) model, contrary to the predictions of perturbation theory.