AI 中文总结
研究五维空间中复杂邻域随机位点渗流问题，用高效C++代码加速计算，计算多个渗流阈值和分形维数，验证渗流阈值与加权配位数关系，得出相关数据及幂律关系。
AI 中文摘要
本文研究了五维空间中复杂邻域的随机位点渗流问题。给出了经典纽曼 - 齐夫算法的高效C++代码（普通工作划分），计算加速将原本获取本文结果所需的2170年单核计算时间大幅缩短。对于复杂邻域（由多个协调区的位点组成，直至第七协调区），计算了127个渗流阈值，其中120个是首次估计。还计算了七个扩展（紧凑）邻域的分形维数，其平均值约为⟨df⟩≈3.5581(70)。紧凑邻域分形维数的百分比误差在0.26%至1.76%之间。验证了渗流阈值依赖加权配位数ζ的普遍性，表现为幂律（pc∝ζ−g），g≈0.7913(43)。
英文摘要
In this paper, the random site percolation problem in a five-dimensional space for complex neighborhoods is studied. The efficient C++ code (with ordinary work division) of the classical Newman--Ziff algorithm is presented. The obtained speed-up of computations reduces 2170 years of single-core computations -- necessary for obtaining the results presented in this paper -- much below the typical half-decay time of the scientist. For complex neighborhoods, it is for neighborhoods composed with sites taken from several coordination zones (up to the seventh coordination zone), the 127 percolation thresholds are calculated (with 120 among them being estimated for the first time). For seven extended (compact) neighborhoods, the fractal dimensions are also calculated. The mean value of these fractal dimensions, averaged over these seven compact neighborhoods, is estimated as $\langle d_f\rangle\approx 3.5581(70)$. The percentage errors of the values obtained for the fractal dimensions for compact neighborhoods vary from 0.26\% to 1.76\% with respect to the theoretically predicted value based on scaling relations and the most recent estimates of critical exponents for five-dimensional space. The universality of the percolation threshold as dependent on the weighted coordination number $ζ=\sum_i z_i r_i$ (where $z_i$ is the number of sites in the $i$-th coordination zone and $r_i$ is the Euclidean distance from the sites in the $i$-th coordination zone to the central site) is also verified. The latter manifests itself as the power law ($p_c\proptoζ^{-g}$) with $g\approx 0.7913(43)$
Comments9 pages, 4 figures (plus 2 figures in supplemental material), 2 source codes in C++ (in supplemental material)
