The 3d random field Ising model at zero temperature

We study numerically the zero-temperature random field Ising model on cubic lattices of various linear sizes L in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength J. We find that in the infinite volume limit the magnetization is discontinuous in J. The energy and its first J derivative are continuous. The approach to the thermodynamic limit is slow, behaving like $L^{-p}$ with $p \sim 0.8 $ for the Gaussian distribution of the random field. We also study the bimodal distribution $h_{i} = \pm h$, and we find similar results for the magnetization but with a different value of the exponent $p \sim 0.6 $. This raises the question of the validity of universality for the random field problem.