Invaded cluster algorithm for a tricritical point in a diluted Potts model

The invaded cluster approach is extended to the two-dimensional Potts model with annealed vacancies by using the random-cluster representation. Geometrical arguments are used to propose the algorithm which converges to the tricritical point in the two-dimensional parameter space spanned by temperature and the chemical potential of the vacancies. The tricritical point is identified as a simultaneous onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of the “geometrical disorder cluster.” The location of the tricritical point and the concentration of vacancies for $q=1,2,3$ are found to be in good agreement with the best known results. Scaling properties of the percolating scaling cluster and related critical exponents are also presented.

Figure 1

(Color online) Plot of $⟨t{⟩}_L$ vs $1∕L$ for different values of $q$. Plain lines denote the nonlinear fits to the power law form $⟨t{⟩}_L=⟨t{⟩}_t+b{L}^{-x}$.

Figure 2

(Color online) Double logarithmic plot of $\mid p\left(L\right)-p_t\mid$ vs $L$ for different values of $q$. The best known values for $T_t$ are given in Ref. 6 and values $p_t$ can be calculated to be 0.8284271, 0.820168(1), and 0.807931(6) for $q=1$, 2, 3, respectively.

Figure 3

(Color online) Double logarithmic plot of $\mid z\left(L\right)-z_t\mid$ vs $L$ for different values of $q$. The best known values for $G_t$ are given in Ref. 6 and values $z_t$ can be calculated to be 0.02859548, 0.064220(5), and 0.113695(5) for $q=1$, 2, 3, respectively.

Figure 4

(Color online) Double logarithmic plot of percolating FK cluster size $s_{\max }\left(L\right)$ vs $L$ for different values of $q$.

Figure 5

(Color online) Number of red bonds on the percolating FK cluster vs $L$.