  We investigate the two-dimensional $q=3$ and 4 Potts models with a variable
interaction range by means of Monte Carlo simulations. We locate the phase
transitions for several interaction ranges as expressed by the number $z$ of
equivalent neighbors. For not too large $z$, the transitions fit well in the
universality classes of the short-range Potts models. However, at longer ranges
the transitions become discontinuous. For $q=3$ we locate a tricritical point
separating the continuous and discontinuous transitions near $z=80$, and a
critical fixed point between $z=8$ and 12. For $q=4$ the transition becomes
discontinuous for $z > 16$. The scaling behavior of the $q=4$ model with $z=16$
approximates that of the $q=4$ merged critical-tricritical fixed point
predicted by the renormalization scenario.

Statistical Mechanics