Correction-to-scaling exponent for percolation and the Fortuin--Kasteleyn Potts model in two dimensions

Abstract

The number $n_s$ of clusters (per site) of size $s$, a central quantity in percolation theory, displays at criticality an algebraic scaling behavior of the form $n_s\simeq s^{-\tau }A(1+B{s}^{-\Omega })$. For the Fortuin--Kasteleyn representation of the $Q$-state Potts model in two dimensions, the Fisher exponent $\tau$ is known as a function of the real parameter $0\le Q\le 4$, and, for bond percolation (the $Q\to 1$ limit), the correction-to-scaling exponent is derived as $\Omega =72/91$. We theoretically derive the exact formula for the correction-to-scaling exponent $\Omega =8/[(2g+1)(2g+3)]$ as a function of the Coulomb-gas coupling strength $g$, which is related to $Q$ by $Q=2+2\cos (2\pi g)$. Using an efficient Monte Carlo cluster algorithm, we study the O($n$) loop model on the hexagonal lattice, which is in the same universality class as the $Q=n^2$ Potts model, and has significantly suppressed finite-size corrections and critical slowing-down. The predictions of the above formula include the exact value for percolation as a special case and agree well with the numerical estimates of $\Omega$ for both the critical and tricritical branches of the Potts model.

arXiv commentThe document contains the paper (pdflatex, 15 pages), and 6 pdf figures. Minor changes with respect v1. Final version for publication

References in corpus (14)

• Worm algorithms for classical statistical models
• Universal Amplitude Ratios in the Critical Two-Dimensional Ising Model on a Torus
• Logarithmic Corrections and Finite-Size Scaling in the Two-Dimensional 4-State Potts Model
• Operator Spectrum and Exact Exponents of the Fully Packed Loop Model
• Geometrical vs. Fortuin-Kasteleyn Clusters in the Two-Dimensional $q$-State Potts Model
• Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice
• Correction-to-scaling exponent for two-dimensional percolation
• Cluster simulations of loop models on two-dimensional lattices
• Fractal geometry of critical Potts clusters
• The O(n) model on the annulus
• Worm Monte Carlo study of the honeycomb-lattice loop model
• Dynamic critical behavior of the Chayes-Machta-Swendsen-Wang algorithm
• Geometric properties of two-dimensional O(n) loop configurations
• Backbone and shortest-path exponents of the two-dimensional $Q$-state Potts model

Cited by in corpus (3)

• Finite-size corrections from the subleading magnetic scaling field for the Ising and Potts models in two dimensions
• Backbone three-point correlation function in the two-dimensional Potts model
• Extended-range percolation in five dimensions