Bridges in the random-cluster model

TL;DR

This paper investigates the stability and fragility of clusters in the random-cluster model, deriving exact relations and uncovering behaviors related to bridges, with implications for understanding critical phenomena and cluster structures.

Contribution

It introduces a classification of edges based on relevance to connectivity and derives exact relations for the finite-size scaling of bridges and non-bridges in the model.

Findings

Characterizes the point of maximum cluster fragility linked to bridge load.

Shows divergence of bridge variance below a specific coupling value in 2D.

Provides improved estimates of backbone fractal dimension through bridge pruning.

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.