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 $n_s\simeq s^{-\ddot{I}„}A(1+B{s}^{-\hat{I}\text{c◯}})$. For the Fortuin--Kasteleyn representation of the $Q$-state Potts model in two dimensions, the Fisher exponent $\ddot{I}„$ 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 $\hat{I}\text{c◯}=72/91$. We theoretically derive the exact formula for the correction-to-scaling exponent $\hat{I}\text{c◯}=8/[(2g+1)(2g+3)]$ as a function of the Coulomb-gas coupling strength $g$, which is related to $Q$ by $Q=2+2\cos (2\ddot{I}€g)$. 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 $\hat{I}\text{c◯}$ for both the critical and tricritical branches of the Potts model.

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