  In this note, we clarify the stability of the large-N functional RG fixed
points of the order/disorder transition in the random-field (RF) and
random-anisotropy (RA) O(N) models. We carefully distinguish between infinite
N, and large but finite N. For infinite N, the Schwarz-Soffer inequality does
not give a useful bound, and all fixed points found in cond-mat/0510344 (Phys.
Rev. Lett. 96, 197202 (2006)) correspond to physical disorder. For large but
finite N (i.e. to first order in 1/N) the non-analytic RF fixed point becomes
unstable, and the disorder flows to an analytic fixed point characterized by
dimensional reduction. However, for random anisotropy the fixed point remains
non-analytic (i.e. exhibits a cusp) and is stable in the 1/N expansion, while
the corresponding dimensional-reduction fixed point is unstable. In this case
the Schwarz-Soffer inequality does not constrain the 2-point spin correlation.
We compute the critical exponents of this new fixed point in a series in 1/N
and to 2-loop order.
