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 an 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.

Figure 1

The amplitude ratio $r_s$ for bond percolation (left panel), a log-log plot of the difference between consecutive ratios (middle panel), and the local exponent of the difference plot (right panel).

Figure 2

Estimates of the amplitude ratio ${\Gamma }^{-}∕{\Gamma }^{+}$ for bond percolation from fits to the form (6) with $\alpha =0.85$.

Figure 3

The high-density amplitude $\tilde{\Gamma }=q_c^{-4}{\Gamma }^{+}$ for site percolation.