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 (3D) = 0.221 654 631(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) dC(1)= (P) over tilde (x) dx with x C-1/ L-dF (L is the linear size), where the fractal dimension d(F) is identical to the magnetic exponent. An interesting bimodal feature is observed in distribution (P) over tilde (x) in three dimensions, and attributed to the different approaching behaviors for K -> K-c+0(+/-). To characterize the compactness of the FK clusters, we measure their graph distances and determine the shortest-path exponents as d(min) (3D) = 1.259 36(12) and d(min) (2D) = 1.094 0(2). Further, by excluding all the bridges from the occupied bonds, we obtain bridge-free configurations and determine the backbone exponents as d(B) (3D) = 2.167 3(15) and d(B) (2D) = 1.732 1(4). The estimates of the universal wrapping probabilities for the 3D Ising model and of the geometric critical exponents d(min) and d(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.