Confirming universality of the fractal dimension of incipient percolation cluster for complex neighborhoods.

In this paper, a 40-year-old theorem is tested, that the incipient percolation cluster has a fractal dimension equal to 91/48. With the Newman-Ziff algorithm, we measure the mass M of the incipient percolation cluster (i.e., the size of the largest cluster at the percolation threshold) versus the linear system size L which (after averaging over [Formula: see text] system realizations) nicely follows the power law [Formula: see text] with exponents [Formula: see text] ranging from 1.893954 to 1.89823 for the square lattice. The obtained fractal dimension agrees well with its analytical partner and those confirmed numerically earlier for compact neighborhoods with the nearest-neighbors on triangular and square lattices and holds for other considered neighborhoods on square lattice, including those that are not-compact. With six digits of the accuracy of reaching the percolation threshold, the percentage error of the numerical values obtained for the fractal dimension ranges from 0.028‰ to 1.264‰, which strengthens the earlier results confirming the universality of the fractality of the incipient percolation cluster. Using the Hoshen-Kopelman algorithm for cluster identification for [Formula: see text] and the box-counting procedure for the evaluation of the fractal dimension, after [Formula: see text] system realizations, we reached the percentage error of the numerical values obtained for the fractal dimension from 5‰ to 7‰, which is much worse than the percentage error obtained directly from the mass of the incipient percolation cluster as a function of the linear size of the system. Our results indicate that universality of fractality of the incipient percolation cluster is valid also for complex (non-compact) neighborhoods, which allow for occupied site connections with more 'holes' in cluster than allowed for extended (compact) neighborhoods. Also for a simple cubic lattice we get [Formula: see text]-independently on assumed neighborhoods-although these values are slightly higher than known in literature.