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