Percolation effects in the Fortuin-Kasteleyn 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.

arXiv comment9 pages, 12 figures

References in corpus (10)

• The Random-Cluster Model
• Critical random graphs: Diameter and mixing time
• Kinetic Theory of Random Graphs: from Paths to Cycles
• Finite-size scaling above the upper critical dimension
• Random-length Random Walks and Finite-size scaling in high dimensions
• Geometric explanation of anomalous finite-size scaling in high dimensions
• Critical percolation clusters in seven dimensions and on a complete graph
• Geometric properties of the Fortuin-Kasteleyn representation of the Ising model
• Complete graph and Gaussian fixed point asymptotics in the five-dimensional Fortuin-Kasteleyn Ising model with periodic boundaries
• Complete graph asymptotics for the Ising and random cluster models on 5D grids with cyclic boundary

Cited by in corpus (7)

• Geometric Upper Critical Dimensions of the Ising Model
• Geometric scaling behaviors of the Fortuin-Kasteleyn Ising model in high dimensions
• Geometric properties of the complete-graph Ising model in the loop representation
• Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation
• Logarithmic Finite-Size Scaling of the Four-Dimensional Ising Model
• Percolation in the two-dimensional Ising model
• Universality of the complete-graph Potts model with $0<q\le 2$