  The random field Ising model (RFIM) is central to the study of disordered
systems. Yet, for a long time it eluded realization in ferromagnetic systems
because of the difficulty to produce locally random magnetic fields. Recently
it was shown that in anisotropic dipolar magnetic insulators, the archetypal of
which is the $LiHo_xY_{1-x}F_4$ system, the RFIM can be realized in both
ferromagnetic and spin glass phases. The interplay between an applied
transverse field and the offdiagonal terms of the dipolar interaction produce
effective longitudinal fields, which are random in sign and magnitude as a
result of spatial dilution. In this paper we use exact numerical
diagonalization of the full Hamiltonian of Ho pairs in $LiHo_xY_{1-x}F_4$ to
calculate the effective longitudinal field beyond the perturbative regime. In
particular, we find that nearby spins can experience an effective field larger
than the intrinsic dipolar broadening (of quantum states in zero field) which
can therefore be evidenced in experiments. We then calculate the magnetization
and susceptibility under several experimental protocols, and show how these
protocols can produce direct measurement of the effective longitudinal field.
