  We calculate magnetic couplings in the $J_{pd}$ model for dilute magnets, in
order both to identify the relevant parameters which control ferromagnetism and
also to bridge the gap between first principle calculations and model
approaches. The magnetic exchange interactions are calculated
non-perturbatively and disorder in the configuration of impurities is treated
exacly, allowing us to test the validity of effective medium theories.
  Results differ qualitatively from those of weak coupling. In contrast to mean
field theory, increasing $J_{pd}$ may not favor high Curie temperatures: $T_C$
scales primarily with the bandwidth. High temperature ferromagnetism at small
dilutions is associated with resonant structure in the p-band. Comparison to
diluted magnetic semiconductors indicate that Ga(Mn)As has such a resonant
structure and thus this material is already close to optimality.
