Large clusters in supercritical percolation.

Abstract

The statistical behavior of the size of large finite clusters in supercritical percolation on a finite lattice is investigated (below the critical dimension of the space d(c)=6). For this purpose, an approximate system of ordinary differential equations for a number of finite clusters is obtained. The correlation between the critical exponents zeta that determine the cluster decay law (ln n(s) approximately -s(zeta)) and the surface of clusters is shown. It is found that for clusters without self-intersections having a maximal surface zeta=1. For clusters with a small number of self-intersections zeta=1-eta. Here eta is a function depending on the ratio of the surface area of a cluster to its size, which tends to zero, when the surface tends to a maximum. For compact clusters with a minimum or near-minimum surface area, the first correction to the cluster decay law above percolation threshold (ln n(s) approximately -s((d-1)/d)) has been found on the basis of the drop model and the derived system of equations. The predictions are tested numerically on two- and three-dimensional lattices by Monte Carlo simulations. The results of the work allow one to conclude that above the percolation threshold majority of large clusters are compact and that the cluster surface is the main factor affecting its behavior in supercritical percolation.