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 prove 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 (normalized) variance of the number of (non-)bridges, showing that it diverges in two dimensions below the value 4cos2(π/3)=0.2315891⋯ of the cluster coupling q. Finally, we show 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.
Highlights
Percolation is probably the most widely discussed and arguably the simplest model of critical phenomena
There, the scaling limit of critical percolation can be related to the Coulomb gas [4] and conformal field theory [5], leading to exact results for most critical exponents and certain correlation functions
Smirnov and co-workers used the concept of discrete analyticity to establish rigorously that the scaling limits of critical percolation [7,8] and the Ising model [9] on the triangular lattice are conformally invariant, and cluster boundaries in these models converge to certain classes of SLE traces
Summary
Percolation is probably the most widely discussed and arguably the simplest model of critical phenomena. Removing singly connected bonds, which we call bridges, generates an additional connected component, while this is not the case for multiply connected bonds or non-bridges This separation allows us to control the effect of a local edge manipulation (open ↔ closed) on the weight of the configuration as in Eq (1.1). By definition, the removal of a bridge leads to the generation of a new component and the breakup of an existing cluster, understanding the properties of bridge bonds is crucial to the understanding of fragmentation phenomena in the framework of a lattice model This connection was investigated in our recent letter [20], where we studied the fragmentation rate and kernel, and related the associated scaling exponents to the more standard critical exponents and fractal dimensions.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
