Some geometric critical exponents for percolation and the random-cluster model

Abstract

We introduce several infinite families of new critical exponents for the random-cluster model and present scaling arguments relating them to the k-arm exponents. We then present Monte Carlo simulations confirming these predictions. These new exponents provide a convenient way to determine k-arm exponents from Monte Carlo simulations. An understanding of these exponents also leads to a radically improved implementation of the Sweeny Monte Carlo algorithm. In addition, our Monte Carlo data allow us to conjecture an exact expression for the shortest-path fractal dimension d_min in two dimensions: d_min = (g+2)(g+18)/(32g) where g is the Coulomb-gas coupling, related to the cluster fugacity q via q = 2 + 2 cos(gπ/2) with 2 \le g \le 4.

LaTeX2e/Revtex4. Version 2 is completely rewritten to make the exposition more reader-friendly; it consists of a 4-page main paper (including 3 figures) and a 2-page EPAPS appendix (given as a single Postscript file). To appear in Phys Rev E

References in corpus (3)

• A Guide to Stochastic Loewner Evolution and its Applications
• Critical speeding-up in a local dynamics for the random-cluster model
• Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm

Cited by in corpus (22)

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• Worm Monte Carlo study of the honeycomb-lattice loop model
• The O(n) loop model on a three-dimensional lattice
• Geometric structure of percolation clusters
• Shortest path and Schramm-Loewner Evolution
• Crossover from Isotropic to Directed Percolation
• Geometric properties of the Fortuin-Kasteleyn representation of the Ising model
• Bulk and boundary critical behaviour of thin and thick domain walls in the two-dimensional Potts model
• Efficient simulation of the random-cluster model
• Fragmentation of fractal random structures
• Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model
• Conductivity of Coniglio-Klein clusters
• Active spanning trees and Schramm-Loewner evolution
• Bridges in the random-cluster model
• Percolation of the Site Random-Cluster Model by Monte Carlo Method
• Dynamic connectivity algorithms for Monte Carlo simulations of the random-cluster model
• Critical site percolation on the triangular lattice: From integrability to conformal partition functions
• Ising-like phase transition of an n-component Eulerian face-cubic model
• Sweeny dynamics for the random-cluster model with small $Q$
• Universality of the complete-graph Potts model with $0<q\le 2$
• Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition
• An upper bound on geodesic length in 2D critical first-passage percolation