Percolation effects in the Fortuin-Kasteleyn Ising model on the complete graph

TL;DR

This study uses extensive simulations to analyze percolation phenomena in the FK representation of the mean-field Ising model on a complete graph, revealing percolation effects and finite-size scaling behaviors.

Contribution

It provides the first numerical evidence of percolation effects in the FK Ising model on a complete graph, highlighting the influence of an asymptotically vanishing configuration space sector.

Findings

Configuration space contains a vanishing sector with uncorrelated percolation scaling.

Cluster-size distribution follows Fisher exponent of uncorrelated percolation.

Percolation effects are demonstrated in the FK Ising model on the complete graph.

Abstract

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 behaviour 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.