  The random-field Ising model is one of the few disordered systems where the
perturbative renormalization group can be carried out to all orders of
perturbation theory. This analysis predicts dimensional reduction, i.e., that
the critical properties of the random-field Ising model in $D$ dimensions are
identical to those of the pure Ising ferromagnet in $D-2$ dimensions. It is
well known that dimensional reduction is not true in three dimensions, thus
invalidating the perturbative renormalization group prediction. Here, we report
high-precision numerical simulations of the 5D random-field Ising model at zero
temperature. We illustrate universality by comparing different probability
distributions for the random fields. We compute all the relevant critical
exponents (including the critical slowing down exponent for the ground-state
finding algorithm), as well as several other renormalization-group invariants.
The estimated values of the critical exponents of the 5D random-field Ising
model are statistically compatible to those of the pure 3D Ising ferromagnet.
These results support the restoration of dimensional reduction at $D = 5$. We
thus conclude that the failure of the perturbative renormalization group is a
low-dimensional phenomenon. We close our contribution by comparing universal
quantities for the random-field problem at dimensions $3 \leq D < 6$ to their
values in the pure Ising model at $D-2$ dimensions and we provide a clear
verification of the Rushbrooke equality at all studied dimensions.
