Abstract
Low concentration series are generated for moments of the percolation cluster size distribution, Gamma j=(sj-1) (s is the number of sites on a cluster) for j=2, . . ., 8 and general dimensionality d. These diverge at pc as Gamma j approximately Aj(pc-p)- gamma j with gamma j= gamma j= gamma +(j-2) Delta , where delta = gamma + beta is the gap exponent. The series yield new accurate values for Delta and beta , Delta =2.23+or-0.05, 2.10+or-0.04, 2.03+or-0.05 and beta =0.44+or-0.15, 0.66+or-0.09, 0.83+or-0.08 at d=3, 4, 5. In addition, ratios of the form AjAk/AmAn, with j+k=m+n, are shown to be universal. New values for some of these ratios are evaluated from the series, from the epsilon expansion ( epsilon =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.
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