  We show that, contrary to previous suggestions based on computer simulations
or erroneous theoretical treatments, the critical points of the random-field
Ising model out of equilibrium, when quasi-statically changing the applied
source at zero temperature, and in equilibrium are not in the same universality
class below some critical dimension $d_{DR}\approx 5.1$. We demonstrate this by
implementing a non-perturbative functional renormalization group for the
associated dynamical field theory. Above $d_{DR}$, the avalanches, which
characterize the evolution of the system at zero temperature, become irrelevant
at large distance, and hysteresis and equilibrium critical points are then
controlled by the same fixed point. We explain how to use computer simulation
and finite-size scaling to check the correspondence between in and out of
equilibrium criticality in a far less ambiguous way than done so far.
