We have studied spin-spin correlation functions in the ordered phase of the two-dimensional q -state Potts model with q = 10, 15, and 20 at the first-order transition point β t . Through extensive Monte Carlo simulations we obtain strong numerical evidence that the correlation length in the ordered phase agrees with the exactly known and recently numerically confirmed correlation length in the disordered phase: ξ o (β t ) = ξ d (β t ). As a by-product we find the energy moments in the ordered phase at β t in very good agreement with a recent large q -expansion.
We report numerical simulations of two-dimensional
q-state Potts models with emphasis on a new quantity for the computation of spatial correlation lengths. This quantity is the cluster-diameter distribution function
Gdiam(x), which measures the distribution of the diameter of stochastically defined cluster. Theoretically it is predicted to fall off exponentially for large diameter
x, ${G}^{\mathrm{diam}}\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}x/\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the correlation length as usually defined through the large-distance behavior of two-point correlation functions. The results of our extensive Monte Carlo study in the disordered phase of the models with
q=10, 15, and
20 on large square lattices of size $300\ifmmode\times\else\texttimes\fi{}300$, $120\ifmmode\times\else\texttimes\fi{}120$, and $80\ifmmode\times\else\texttimes\fi{}80$, respectively, clearly confirm the theoretically predicted behavior. Moreover, using this observable we are able to verify an exact formula for the correlation length ${\ensuremath{\xi}}_{d}({\ensuremath{\beta}}_{t})$ in the disordered phase at the first-order transition point ${\ensuremath{\beta}}_{t}$ with an accuracy of about
1--
2 for all considered values of
q. This is a considerable improvement over estimates derived from the large-distance behavior of standard (projected) two-point correlation functions, which are also discussed for comparison.
Monte Carlo results are presented for a variety of one-dimensional dynamical
q-state Potts models. Our calculations confirm the expected universal value
z=2 for the dynamic scaling exponent. Our results also indicate that an increase in
q at fixed correlation length drives the dynamics into the scaling regime
We consider the exact correlation length calculations for the two-dimensional Potts model at the transition point βt by Klümper, Schadschneider and Zittartz, and by Buffenoir and Wallon. We argue that the correlation length calculated by the latter authors is the correlation length in the disordered phase and then combine their result with duality and the assumption of complete wetting to give an explicit formula for the order-disorder interface tension σod of this model. The result is used to clarify a controversy stemming from different numerical simulations of σod.