Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm

Abstract

A new approach for the determination of the critical percolation concentration, percolation probabilities, and cluster size distributions is presented for the site percolation problem. The novel "cluster multiple labeling technique" is described for both two- and three-dimensional crystal structures. Its distinctive feature is the assignment of alternate labels to sites belonging to the same cluster. These sites are members of a simulated finite random lattice. An algorithm useful for the determination of the critical percolation concentration of a finite lattice is also presented. This algorithm is especially useful when applied in conjunction with the cluster multiple labeling technique. The basic features of this technique are illustrated by applying it to a small planar square lattice. Numerical results are given for a triangular subcrystal containing up to 9 000 000 sites. These results compare favorably with the exact value of the infinite lattice critical percolation concentration.