Ferromagnetic Potts models with multisite interaction

We study the $q$ states Potts model with four site interaction on the square lattice. Based on the asymptotic behaviour of lattice animals, it is argued that when $q\leq 4$ 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 behaviour.