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

Abstract

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 ${K}_{c}(3\mathrm{D})=0.221\phantom{\rule{0.16em}{0ex}}654\phantom{\rule{0.16em}{0ex}}631(8)$. 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 $P({C}_{1},L)\phantom{\rule{0.16em}{0ex}}d{C}_{1}=\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}(x)\phantom{\rule{0.16em}{0ex}}dx$ with $x\ensuremath{\equiv}{C}_{1}/{L}^{{d}_{\mathrm{F}}}$ ($L$ 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 $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{P}(x)$ in three dimensions, and attributed to the different approaching behaviors for $K\ensuremath{\rightarrow}{K}_{c}+{0}^{\ifmmode\pm\else\textpm\fi{}}$. To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as ${d}_{\mathrm{min}}(3\mathrm{D})=1.259\phantom{\rule{0.16em}{0ex}}36(12)$ and ${d}_{\mathrm{min}}(2\mathrm{D})=1.094\phantom{\rule{0.16em}{0ex}}0(2)$. Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as ${d}_{\mathrm{B}}(3\mathrm{D})=2.167\phantom{\rule{0.16em}{0ex}}3(15)$ and ${d}_{\mathrm{B}}(2\mathrm{D})=1.732\phantom{\rule{0.16em}{0ex}}1(4)$. The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents ${d}_{\mathrm{min}}$ 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.