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-centred cubic (f.c.c.), and body-centred cubic (b.c.c.) lattices. Systems with were studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The excess number of clusters per unit length was confirmed to be a universal quantity with a value . Likewise, the critical crossing probability in the direction, with periodic boundary conditions in the plane, was found to follow a universal exponential decay as a function of 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 , . We also report the value for site percolation.