  The phase transitions of random-field q-state Potts models in d=3 dimensions
are studied by renormalization-group theory by exact solution of a hierarchical
lattice and, equivalently, approximate Migdal-Kadanoff solutions of a cubic
lattice. The recursion, under rescaling, of coupled random-field and
random-bond (induced under rescaling by random fields) coupled probability
distributions is followed to obtain phase diagrams. Unlike the Ising model
(q=2), several types of random fields can be defined for q >= 3 Potts models,
including random-axis favored, random-axis disfavored, random-axis randomly
favored or disfavored cases, all of which are studied. Quantitatively very
similar phase diagrams are obtained, for a given q for the three types of field
randomness, with the low-temperature ordered phase persisting, increasingly as
temperature is lowered, up to random-field threshold in d=3, which is
calculated for all temperatures below the zero-field critical temperature.
Phase diagrams thus obtained are compared as a function of $q$. The ordered
phase in the low-q models reaches higher temperatures, while in the high-q
models it reaches higher random fields. This renormalization-group calculation
result is physically explained.
