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

TL;DR

This study uses extensive Monte Carlo simulations to demonstrate the universality of the excess number of clusters and crossing probability function in three-dimensional percolation across different lattice types, and provides precise critical thresholds.

Contribution

It confirms the universality of the excess number of clusters and crossing probability in 3D percolation and determines new precise critical thresholds for f.c.c. and b.c.c. lattices.

Findings

Excess number of clusters $ ilde{b} \,\approx\, 0.412$ is universal.

Crossing probability decays exponentially with L'/L ratio.

New precise critical thresholds for f.c.c. and b.c.c. lattices.

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 $p_c(f.c.c.)= 0.199 236 5 \pm 0.000 001 0$, $p_c(b.c.c.)= 0.245 961 5\pm 0.000 001 0$.