Abstract
We investigate the connectivity properties of the random-cluster model mediated by bridge bonds that, if removed, lead to the generation of new connected components. We study numerically the density of bridges and the fragmentation kernel, i.e., the relative sizes of the generated fragments, and find that these quantities follow a scaling description. The corresponding scaling exponents are related to well known equilibrium critical exponents of the model. Using the Russo-Margulis formalism, we derive an exact relation between the expected density of bridges and the number of active edges. The same approach allows us to study the fluctuations in the numbers of bridges, thereby uncovering a new singularity in the random- cluster model as q < 4 cos2 (π/√3) in two dimensions. For numerical simulations of the model directly in the language of individual bonds, known as Sweeny's algorithm, the prevalence of bridges and the scaling of the sizes of clusters connected by bridges and candidate-bridges play a pivotal role. We discuss several different implementations of the necessary connectivity algorithms and assess their relative performance.
Highlights
The random-cluster model was introduced by Fortuin and Kasteleyn as a generalization of the much studied Ising model of ferromagnetism and the percolation model introduced in the 1950s [1]
We find these values to be such that the advantage of the single-bond dynamics is destroyed for the sequential breadth-first search (SBFS), interleaved breadth-first search (IBFS) and union-and-find algorithm (UF) approaches apart from, maybe, in the vicinity of the tricritical point q = 4
We have presented a general implementation of Sweeny’s algorithm for the random cluster model that works independent of the structure and dimensionality of the lattice and features, at least asymptotically, an improved efficiency in decorrelating the system as compared to the better known Swendsen-Wang-Chayes-Machta algorithm
Summary
The random-cluster model was introduced by Fortuin and Kasteleyn as a generalization of the much studied Ising model of ferromagnetism and the percolation model introduced in the 1950s [1]. Simulation using Sweeny’s algorithm While the construction of Swendsen and Wang [5], published in 1987, of a rejection-free update of spin clusters in connection with bond variables implied by the random-cluster representation is well known and widely used to simulate Ising, Potts and (through the cluster embedding trick) continuous spin models [7], a simpler and more direct approach was proposed by Sweeny [8] in 1983 already His approach is rooted in the results of Fortuin and Kasteleyn [1] in directly attempting to sample bond configurations of the RC model according to the weight wRC(A) = qK(A)v|A|,. For simulations of the RC model, we can perform each operation in a runtime asymptotically proportional to log L, clearly outperforming the other approaches at criticality, cf. Tab. 1
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