  We study the long-distance behavior of the O(N) model in the presence of
random fields and random anisotropies correlated as ~1/x^{d-sigma} for large
separation x using the functional renormalization group. We compute the fixed
points and analyze their regions of stability within a double epsilon=d-4 and
sigma expansion. We find that the long-range disorder correlator remains
analytic but generates short-range disorder whose correlator develops the usual
cusp. This allows us to obtain the phase diagrams in (d,sigma,N) parameter
space and compute the critical exponents to first order in epsilon and sigma.
We show that the standard renormalization group methods with a finite number of
couplings used in previous studies of systems with long-range correlated random
fields fail to capture all critical properties. We argue that our results may
be relevant to the behavior of He-3A in aerogel.
