The Universal Amplitude Ratio Gamma⁻/Gamma⁺ for Two-Dimensional Percolation

The amplitude ratio of the susceptibility (or second size-moment) for two-dimensional percolation is calculated by two series methods and also by Monte-Carlo simulation. The first series method is a new approach based upon integrating approximations to the scaling function. The second series method directly uses low- and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value $\Gamma^{-}/\Gamma^{+} = 162.5 \pm 2$, a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.