  We derive an effective Hamiltonian for ${\rm Ga}_{1-x}{\rm Mn}_x {\rm As}$ in
the dilute limit, where ${\rm Ga}_{1-x}{\rm Mn}_x {\rm As}$ can be described in
terms of spin $F=3/2$ polarons hopping between the {\rm Mn} sites and coupled
to the local {\rm Mn} spins. We determine the parameters of our model from
microscopic calculations using both a variational method and an exact
diagonalization within the so-called spherical approximation. Our approach
treats the extremely large Coulomb interaction in a non-perturbative way, and
captures the effects of strong spin-orbit coupling and Mn positional disorder.
We study the effective Hamiltonian in a mean field and variational calculation,
including the effects of interactions between the holes at both zero and finite
temperature. We study the resulting magnetic properties, such as the
magnetization and spin disorder manifest in the generically non-collinear
magnetic state. We find a well formed impurity band fairly well separated from
the valence band up to $x_{\rm active} \lesssim 0.015$ for which finite size
scaling studies of the participation ratios indicate a localization transition,
even in the presence of strong on-site interactions, where $x_{\rm
active}<x_{\rm nom}$ is the fraction of magnetically active Mn. We study the
localization transition as a function of hole concentration, Mn positional
disorder, and interaction strength between the holes.
