  By performing a high-statistics simulation of the $D=4$ random-field Ising
model at zero temperature for different shapes of the random-field
distribution, we show that the model is ruled by a single universality class.
We compute to a high accuracy the complete set of critical exponents for this
class, including the correction-to-scaling exponent. Our results indicate that
in four dimensions: (i) dimensional reduction as predicted by the perturbative
renormalization group does not hold and (ii) three independent critical
exponents are needed to described the transition.
