A Monte Carlo algorithm has been used to study the cubic ferromagnet w
ith a random uniaxial single-site anisotropy on simple cubic lattices,
in the strong anisotropy limit. For both two-component and three-comp
onent spins the phase diagram contains a ''domain-type'' ferromagnetic
phase for strong random anisotropy, in addition to the simple ferroma
gnetic phase which is seen for weak random anisotropy. For two-compone
nt spins both paramagnet-ferromagnet transitions seem to be second ord
er. For three-component spins the magnetization probably disappears di
scontinuously, and there appears to be a thin layer of infinite suscep
tibility phase between the paramagnetic and ferromagnetic phases.