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We consider the zero-temperature fixed points controlling the criticalbehavior 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 theregion of the $(N,d)$ plane where one passes from a critical behaviorsatisfying the $d\rightarrow d-2$ dimensional reduction to one where it breaksdown due to the appearance of strong enough nonanalyticities in the functionaldependence of the cumulants of the renormalized disorder. We unveil anintricate and unusual behavior.

Fixed points and their stability in the functional renormalization group of random field models