Geometric properties of two-dimensional critical and tricritical Potts models

We investigate geometric properties of the general q-state Potts model in two dimensions, and define geometric clusters as sets of lattice sites in the same Potts state, connected by nearest-neighbor bonds with variable probability p. We find that, besides the random-cluster fixed point, both the critical and the tricritical Potts models have another fixed point in the p direction. For the critical model, the random-cluster fixed point p(r) is unstable and the other point p(g) > or =p(r) is stable; while p(r) is stable and p(g) < or =p(r) is unstable at tricriticality. Moreover, we show that the fixed point p(g) of a critical and tricritical q-state Potts models can be regarded to correspond to p(r) of a tricritical and critical q'-state Potts models, respectively. In terms of the coupling constant of the Coulomb gas g, these two models are related as gg'=16. By means of Monte Carlo simulations, we obtain p(g)=0.6227(2) and 0.6395(2) for the tricritical Blume-Capel and the q=3 Potts model, respectively, and confirm the predicted values of the magnetic and bond-dilution exponents near p(g).