Universality of the excess number of clusters and the crossing probability function in three-dimensional percolation

Extensive Monte-Carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three-dimensional percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and bodycentered cubic (b.c.c.) lattices. Systems L × L × L′ with L′ >> L were studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The excess number of clusters b̃ per unit length was confirmed to be a universal quantity with a value b̃ ≈ 0.412. Likewise, the critical crossing probability in the L ′ direction, with periodic boundary conditions in the L × L plane, was found to follow a universal exponential decay as a function of r = L ′ /L for large r. Simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f.c.c. and b.c.c. lattices, yielding pc(f.c.c.) = 0.199 236 5±0.000 001 0, pc(b.c.c.) = 0.245 961 5±0.000 001 0. We also report the value pc(s.c.) = 0.311 608 0 ± 0.000 000 4 for site percolation. PACS numbers(s): 64.60Ak, 05.70.Jk Typeset using REVTEX 1