Potts models in random fields

The q-state Potts model is studied in the presence of random fields, which locally prefer ordering of any one of the q states. In d dimensions, the transition is expected to become first-order for q>qc(d). As in the nonrandom case, mean-field theory still yields qc(d)=2 for all d. Fluctuations are argued to shift the nonrandom value, qc0(d), into a significantly higher value, qc(d). For qc0(d)<q<qc(d) we thus expect random fields to turn the discontinuous transitions into continuous ones. At d=3 this probably includes the experimentally realizable cases q=3 and 4.