  We study how universality classes of O(N)-symmetric models depend
continuously on the dimension d and the number of field components N. We
observe, from a renormalization group perspective, how the implications of the
Mermin-Wagner-Hohenberg theorem set in as we gradually deform theory space
towards d=2. For fractal dimension in the range 2<d<3 we observe, for any N
bigger than or equal to 1, a finite family of multi-critical effective
potentials of increasing order. Apart for the N=1 case, these disappear in d=2
consistently with the Mermin-Wagner-Hohenberg theorem. Finally, we study
O(N=0)-universality classes and find an infinite family of these in two
dimensions.
