  We consider the zero-temperature fixed points controlling the critical
behavior of the $d$-dimensional random-field Ising, and more generally $O(N)$,
models. We clarify the nature of these fixed points and their stability in the
region of the $(N,d)$ plane where one passes from a critical behavior
satisfying the $d\rightarrow d-2$ dimensional reduction to one where it breaks
down due to the appearance of strong enough nonanalyticities in the functional
dependence of the cumulants of the renormalized disorder. We unveil an
intricate and unusual behavior.
