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We study the critical behavior and phase diagram of the $d$-dimensionalrandom field O(N) model by means of the nonperturbative functionalrenormalization group approach presented in the preceding paper. We show thatthe dimensional reduction predictions, obtained from conventional perturbationtheory, break down below a critical dimension $d_{DR}(N)$ and we provide adescription of criticality, ferromagnetic ordering and quasi-long range orderin the whole $(N,d)$ plane. Below $d_{DR}(N)$, our formalism gives access toboth the typical behavior of the system, controlled by zero-temperature fixedpoints with a nonanalytic dimensionless effective action, and to the physics ofrare low-energy excitations ("droplets"), described at nonzero temperature bythe rounding of the nonanalyticity in a thermal boundary layer.

physical review. b, condensed matter and materials physics

Non-Perturbative Functional Renormalization Group for Random Field  Models and Related Disordered Systems. II: Results for the Random Field O(N)  Model