Percolation Effects In The Fortuin-Kasteleyn Ising Model On The Complete Graph

The Fortuin-Kasteleyn (FK) random-cluster model, which can be exactly mapped from the q-state Potts spin model, is a correlated bond percolation model. By extensive Monte Carlo simulations, we study the FK bond representation of the critical Ising model (q = 2) on a finite complete graph, i.e., the mean-field Ising model. We provide strong numerical evidence that the configuration space for q = 2 contains an asymptotically vanishing sector in which quantities exhibit the same finite-size scaling as in the critical uncorrelated bond percolation (q = 1) on the complete graph. Moreover, we observe that, in the full configuration space, the power-law behavior of the cluster-size distribution for the FK Ising clusters except the largest one is governed by a Fisher exponent taking the value for q = 1 instead of q = 2. This demonstrates the percolation effects in the FK Ising model on the complete graph.