On random fields, random anisotropies, nonlinear sigma-models, and dimensional reduction.

It is demonstrated that, in contrast to earlier claims, there is no perturbative fixed point of the renormalization group in 4+epsilon dimensions. The flows go into regimes where non-perturbative effects are important and it is argued that dimensional reduction is likely to break down. This is the first example of which we are aware of a problem for which an infinite number of marginal operators play an important role. Various suggestions concerning the behavior of random anisotropy magnets are discussed in light of the present results.