Critical behavior of the anisotropicn-vector model, self-avoiding rings, and polymerization

We study the critical behavior of the m-anisotropic n-vector model with n and m both as continuous variables [0≤n,m<4-O(ε), ε=4-d, d = dimensionality of space] to first order in ε. The limit n→0, $m>0$ of the model is of interest as a model of self-avoiding rings and of polymerization. For n≥m,n,m integers, the critical behavior of the model is known to be that of the isotropic m-vector model, i.e., the $O(m)$ model. Here we prove that the critical behavior of the anisotropic model is always identical with that of the $O(m)$ model for real n,m regardless of whether n≥m or n<m. In particular, we prove that a single self-avoiding ring and a single self-avoiding walk belong to the same universality class of the $O(0)$ model, while polymerization belongs to the universality class of the $O(m)$ model, $m>0$.