Amplitude ratios and β estimates from general dimension percolation moments

Low concentration series are generated for moments of the percolation cluster size distribution, T_j=(s^J-') (s is the number of sites on a cluster) for j = 2,…, 8 and general dimensionality d. These diverge at p_cas (FORMULA PRESENTED.), where (FORMULA PRESENTED.) is the gap exponent. The series yield new accurate values for Δand βΔ = 2.23 ± 0.05, 2.10 ± 0.04, 2.03 ± 0.05 and 946; = 0.44 ± 0.15, 0.66 ± 0.09, 0.83 ± 0.08 at d = 3, 4, 5. In addition, ratios of the form AjA_k/A_mA_n„, with j+ k = m + n, are shown to be universal. New values for some of these ratios are evaluated from the series, from the e expansion (έ = 6 — d) and exactly (in d = 1 and on the Bethe lattice). The results are in excellent agreement with each other for all dimensions. Results for different lattices at d = 2, 3 agree very well. These amplitude ratios are much better behaved than other ratios considered in the past, and should thus be more useful in characterising percolating systems.