Malarz Krzysztof, Krawczyk Malgorzata J
Faculty of Physics and Applied Computer Science, AGH University, al. Mickiewicza 30, 30-059, Kraków, Poland.
Sci Rep. 2025 Sep 25;15(1):32920. doi: 10.1038/s41598-025-17370-x.
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.
在本文中，一个有着40年历史的定理得到了验证，即初始渗流团簇的分形维数等于91/48。使用纽曼 - 齐夫算法，我们测量了初始渗流团簇的质量M（即渗流阈值处最大团簇的大小）与线性系统尺寸L的关系，对于正方形晶格，在对[公式：见正文]个系统实现进行平均后，它很好地遵循幂律[公式：见正文]，指数[公式：见正文]的范围为1.893954至1.89823。所获得的分形维数与其理论值吻合良好，并且与早期在三角形和正方形晶格上具有最近邻的紧凑邻域的数值结果一致，对于正方形晶格上其他考虑的邻域也成立，包括那些不紧凑的邻域。在达到渗流阈值的六位精度下，分形维数的数值百分比误差范围为0.028‰至1.264‰，这强化了早期证实初始渗流团簇分形性普遍性的结果。对于[公式：见正文]，使用霍申 - 科普尔曼算法进行团簇识别，并使用盒计数法评估分形维数，在[公式：见正文]个系统实现之后，我们得到的分形维数数值百分比误差为5‰至7‰，这比直接从初始渗流团簇质量作为系统线性尺寸的函数获得的百分比误差要差得多。我们的结果表明，初始渗流团簇分形性的普遍性对于复杂（非紧凑）邻域也是有效的，这种邻域允许团簇中比扩展（紧凑）邻域有更多“空洞”的占据位点连接。对于简单立方晶格也是如此，我们得到[公式：见正文]，与假设的邻域无关，尽管这些值略高于文献中已知的值。
