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

Abstract

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 body-centered cubic (b.c.c.) lattices. Systems L x L x 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 $\tilde{b}$ per unit length was confirmed to be a universal quantity with a value $\tilde{b}\approx 0.412$. Likewise, the critical crossing probability in the L' direction, with periodic boundary conditions in the L x 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 $p_c(f.c.c.)=0.1992365\pm 0.0000010$, $p_c(b.c.c.)=0.2459615\pm 0.0000010$.

14 pages, 7 figures, LaTeX, submitted to J. Phys. A: Math. Gen, added references, corrected typos

References in corpus (5)

• Precise determination of the bond percolation thresholds and finite-size scaling corrections for the s.c., f.c.c., and b.c.c. lattices
• The Number of Incipient Spanning Clusters in Two-Dimensional Percolation
• Universality of finite-size corrections to the number of critical percolation clusters
• Exact results at the 2-D percolation point
• Effects of boundary conditions on the critical spanning probability

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