DETERMINATION OF THE AVERAGE NUMBER OF CLUSTERS IN SITE PERCOLATION FOR A SQUARE LATTICE

Abstract

This study demonstrates how analytical expressions for estimating the number of clusters in the problem of nodes on a square lattice can be constructed using the summation over all cluster configurations ("lattice animals"). A methodology was developed to derive explicit expressions for the limiting cases of small and large numbers of clusters, enabling a detailed analysis of the system's behavior under different regimes.

To ensure accurate modeling across the entire range of probabilities, a novel approach was proposed to "stitch" these expressions at a specific point. This stitching point is determined as the largest root of an equation derived by equating the respective analytical formulas. It was found that this point corresponds to the percolation threshold estimate . Additionally, it was shown that this estimate is consistent with the known exact value, confirming the validity and effectiveness of the proposed approach.

Explicit analytical expressions for the best approximations of the number of clusters were derived, showing excellent agreement with simulation results for a square lattice of size . An analytical expression for the best estimate of the percolation threshold was found as .

It was observed that the most accurate approximation of the percolation threshold is achieved for small values of . However, it was noted that the direct computation of "lattice animals" for estimating  may be of limited utility unless new ideas or techniques are introduced to improve this approach.

Based on the results obtained in this study, future work could focus on analytically calculating other statistical characteristics of the percolation process. These include properties such as the size of percolation clusters and critical exponents, which would provide deeper insights into the fundamental behavior of the system near the percolation threshold.