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 $q>\mathrm{}{q}_c\mathrm{}\left(d\right)\mathrm{}$. As in the nonrandom case, mean-field theory still yields $q_c\mathrm{}\left(d\right)=2\mathrm{}$ for all $d$. Fluctuations are argued to shift the nonrandom value, $q_c^0\mathrm{}\left(d\right)\mathrm{}$, into a significantly higher value, $q_c\mathrm{}\left(d\right)\mathrm{}$. For $q_c^0\mathrm{}\left(d\right)<q<\mathrm{}{q}_c\mathrm{}\left(d\right)\mathrm{}$ we thus expect random fields to turn the discontinuous transitions into continuous ones. At $d=3\mathrm{}$ this probably includes the experimentally realizable cases $q=3\mathrm{} \mathrm{and} 4$.