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

Abstract

We present a Monte Carlo study of the Fortuin-Kasteleyn (FK) clusters of the Ising model on the square (2D) and simple-cubic (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 $K_c(3\mathrm{D})=0.221654631(8)$. We then study geometric properties of the FK clusters at criticality. It is demonstrated that the distribution of the critical largest-cluster size $C_1$ follows a single-variable function as $P(C_1,L)\mathrm{d}{C}_1=\tilde{P}(x)\mathrm{d}x$ with $x\equiv C_1/L^{d_{\mathrm{F}}}$ ($L$ is the linear size), and that the fractal dimension $d_{\mathrm{F}}$ is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution $\tilde{P}(x)$ in 3D, and attributed to the different approaching behaviors for $K\to K_c+0^{\pm }$. For a critical FK configuration, the cluster number per site $n(s,L)$ of size $s$ is confirmed to obey the standard scaling form $n(s,L)\sim s^{-\tau }\tilde{n}(s/L^{d_{\mathrm{F}}})$, with hyper-scaling relation $\tau =1+d/d_{\mathrm{F}}$ and the spatial dimension $d$. To further characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as $d_{\min }(3\mathrm{D})=1.2594(2)$ and $d_{\min }(2\mathrm{D})=1.0940(3)$. Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as $d_{\mathrm{B}}(3\mathrm{D})=2.1673(15)$ and $d_{\mathrm{B}}(2\mathrm{D})=1.7321(4)$. 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.

12 pages, 10 figures, 11 tables