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Random-field Ising and $O(N)$ models: Theoretical description through the functional renormalization group

We review the theoretical description of the random field Ising and $O(N)$ models obtained from the functional renormalization group, either in its nonperturbative implementation or, in some limits, in perturbative implementations. The approach solves some of the questions concerning the critical behavior of random-field systems that have stayed pending for many years: What is the mechanism for the breakdown of dimensional reduction and the breaking of the underlying supersymmetry below $d=6$? Can one provide a theoretical computation of the critical exponents, including the exponent ψcharacterizing the activated dynamic scaling? Is it possible to theoretically describe collective phenomena such as avalanches and droplets? Is the critical scaling described by 2 or 3 independent exponents? What is the phase behavior of the random-field $O(N)$ model in the whole ($N$, $d$) plane and what is the lower critical dimension of quasi-long range order for $N=2$? Are the equilibrium and out-of-equilibrium critical points of the RFIM in the same universality class?