Nonperturbative functional renormalization group for random field models and related disordered systems.: II.: Results for the random field O(N) model

We study the critical behavior and phase diagram of the d-dimensional random field O(N) model by means of the nonperturbative functional renormalization group approach presented in the preceding paper. We show that the dimensional-reduction predictions, which are obtained from conventional perturbation theory, break down below a critical dimension d(DR)(N) and we provide a description of criticality, ferromagnetic ordering, and quasi-long range order in the whole (N,d) plane. Below d(DR)(N), our formalism gives access to both the typical behavior of the system, which is controlled by zero-temperature fixed points at which the effective action has a nonanalytic field dependence, and to the physics of rare low-energy excitations (droplets), which is described at nonzero temperature by the rounding of the nonanalyticity in a thermal boundary layer.