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

Area of Science:

• Statistical physics
• Complex systems
• Computational physics

Background:

• The Fortuin-Kasteleyn (FK) random-cluster model is a correlated bond percolation model.
• It maps exactly from the q-state Potts spin model, providing a bridge between spin systems and percolation theory.

Purpose of the Study:

• To investigate the FK bond representation of the critical Ising model (q=2) on a complete graph.
• To analyze finite-size scaling and cluster-size distributions within this model.

Main Methods:

• Extensive Monte Carlo simulations were employed.
• The study focused on the mean-field Ising model on a finite complete graph.

Main Results:

• Strong numerical evidence suggests an asymptotically vanishing sector in the q=2 configuration space.
• This sector exhibits finite-size scaling characteristic of uncorrelated bond percolation (q=1).
• The cluster-size distribution's power-law behavior is governed by the Fisher exponent for q=1, not q=2.

Conclusions:

• Percolation effects are demonstrated in the FK Ising model on a complete graph.
• The findings highlight a connection between correlated and uncorrelated percolation phenomena in specific graph structures.