  We have developed a non-perturbative functional renormalization group
approach for the random field O(N) model (RFO(N)M) that allows us to
investigate the ordering transition in any dimension and for any value of N
including the Ising case. We show that the failure of dimensional reduction and
standard perturbation theory is due to the non-analytic nature of the
zero-temperature fixed point controlling the critical behavior, non-analycity
which is associated with the existence of many metastable states. We find that
this non-analycity leads to critical exponents differing from the dimensional
reduction prediction only below a critical dimension d\_c(N)<6, with
d\_c(N=1)>3.
