Bridges in the random-cluster model

Abstract

The random-cluster model, a correlated bond percolation model, unifies a range of important models of statistical mechanics in one description, including independent bond percolation, the Potts model and uniform spanning trees. By introducing a classification of edges based on their relevance to the connectivity we study the stability of clusters in this model. We derive several exact relations for general graphs that allow us to derive unambiguously the finite-size scaling behavior of the density of bridges and non-bridges. For percolation, we are also able to characterize the point for which clusters become maximally fragile and show that it is connected to the concept of the bridge load. Combining our exact treatment with further results from conformal field theory, we uncover a surprising behavior of the variance of the number of (non-)bridges, showing that these diverge in two dimensions below the value $4{\cos }^2(\pi /\sqrt{3})=0.2315891\cdots$ of the cluster coupling $q$. Finally, it is shown that a partial or complete pruning of bridges from clusters enables estimates of the backbone fractal dimension that are much less encumbered by finite-size corrections than more conventional approaches.

final version as published

References in corpus (8)

• Logarithmic observables in critical percolation
• Operator content of the critical Potts model in d dimensions and logarithmic correlations
• Cluster simulations of loop models on two-dimensional lattices
• Geometric structure of percolation clusters
• Short-range correlations in percolation at criticality
• Efficient simulation of the random-cluster model
• Recursive Percolation
• Fragmentation of fractal random structures

Cited by in corpus (5)

• Critical percolation clusters in seven dimensions and on a complete graph
• Geometric properties of the Fortuin-Kasteleyn representation of the Ising model
• Loop-Cluster Coupling and Algorithm for Classical Statistical Models
• Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model
• Sweeny dynamics for the random-cluster model with small $Q$