Geometric properties of the Fortuin-Kasteleyn representation of the Ising model

We present a Monte Carlo study of the geometric properties of Fortuin-Kasteleyn (FK) clusters of the Ising model on square [two-dimensional (2D)] and simple-cubic [three-dimensional (3D)] lattices. The wrapping probability, a dimensionless quantity characterizing the topology of the FK clusters on a torus, is found to suffer from smaller finite-size corrections than the well-known Binder ratio and yields a high-precision critical coupling as $K_c\left(3\mathrm{D}\right)=0.221654631\left(8\right)$. We then study other geometric properties of FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size $C_1$ follows a single-variable function as $P(C_1,L)d{C}_1=\tilde{P}\left(x\right)dx$ with $x\equiv C_1/L^{d_{\mathrm{F}}}$ ($L$ is the linear size), where the fractal dimension $d_{\mathrm{F}}$ is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution $\tilde{P}\left(x\right)$ in three dimensions, and attributed to the different approaching behaviors for $K\to K_c+0^{\pm }$. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as $d_{\min }\left(3\mathrm{D}\right)=1.25936\left(12\right)$ and $d_{\min }\left(2\mathrm{D}\right)=1.0940\left(2\right)$. Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as $d_{\mathrm{B}}\left(3\mathrm{D}\right)=2.1673\left(15\right)$ and $d_{\mathrm{B}}\left(2\mathrm{D}\right)=1.7321\left(4\right)$. The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents $d_{\min }$ and $d_{\mathrm{B}}$ either improve over the existing results or have not been reported yet. We believe that these numerical results would provide a testing ground in the development of further theoretical treatments of the 3D Ising model.

Figure 1

Plots of $R^{\left(x\right)}$ vs $K$ for different system sizes $L$ (a) and $R^{\left(x\right)}(K,L)-b_1{L}^{-0.83}$ vs $L$ for fixed values of $K$ (b) for the 3D Ising model. The value of $b_1=-0.0393$ is taken from Table 1.

Figure 2

Plots of $C_1/L^{d_F}$ vs $L^{-0.83}$ for the critical 3D Ising model.

Figure 3

Probability density distribution of the largest-cluster size for three dimensions (a) and two dimensions (b) at $K_c$, with $x\equiv C_1/L^{d_{\mathrm{F}}}$.

Figure 4

Probability density distribution of the largest-cluster size for three dimensions (a) and two dimensions (b) with $K=K_c\pm \mathrm{\Delta }{L}^{-y_t}$.

Figure 5

Cluster-size distribution for FK clusters for three dimensions (a) and two dimensions (b) at criticality. In both cases, the insets show $s^{\tau }n\left(s\right)$ vs $s/L^{d_{\mathrm{F}}}$.

Figure 6

Plots of $S/L^{d_{\min }}$ vs $L^{-1.8}$ or $L^{-2}$ for the critical 3D (a) or 2D Ising model (b), respectively.

Figure 7

Plots of $C_{1,\mathrm{bf}}/L^{d_{\mathrm{B}}}$ vs $L^{-0.83}$ or vs $L^{-0.67}$ for the critical 3D (a) or 2D (b) Ising model, respectively.

Figure 8

Plot of $g_{ER}^{\left(x\right)}/L^{y_t}-b_1{L}^{-0.83}$ vs $L^{-2}$ for the 3D Ising model, using three different values of $y_t$. The value of $b_1$ is taken from Table 7.

Figure 9

Plots of the cluster number density $(\rho -{\rho }_0)L^{1.413}$ vs $L^{-1.47}$ for the 3D (a) Ising model and $(\rho -{\rho }_0)L$ vs $L^{-1}$ for the 2D (b) Ising model at the critical temperature.

Figure 10

Plots of $\chi$ vs $L^{-0.83}$ at $K_c$ for the 3D Ising model.