Ferromagnetic Potts models with multisite interaction

We study the q-state Potts modelwith four-site interaction on a square lattice. Based on the asymptotic behavior of lattice animals, it is argued that when q <= 4 the system exhibits a second-order phase transition and when q > 4 the transition is first order. The q = 4 model is borderline. We find 1/ln q to be an upper bound on T-c, the exact critical temperature. Using a low-temperature expansion, we show that 1/(theta ln q), where theta > 1 is a q-dependent geometrical term, is an improved upper bound on T-c. In fact, our findings support T-c = 1/(theta ln q). This expression is used to estimate the finite correlation length in first-order transition systems. These results can be extended to other lattices. Our theoretical predictions are confirmed numerically by an extensive study of the four-site interaction model using the Wang-Landau entropic sampling method for q = 3,4,5. In particular, the q = 4 model shows an ambiguous finite-size pseudocritical behavior.