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}≃s^{-τ}A(1+Bs^{-Ω}). For the Fortuin-Kasteleyn representation of the Q-state Potts model in two dimensions, the Fisher exponent τ is known as a function of the real parameter 0≤Q≤4, and, for bond percolation (the Q→1 limit), the correction-to-scaling exponent is derived as Ω=72/91. We derive theoretically the exact formula for the correction-to-scaling exponent Ω=8/[(2g+1)(2g+3)] as a function of the Coulomb-gas coupling strength g, which is related to Q by Q=2+2cos(2πg). 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 Ω for both the critical and tricritical branches of the Potts model.