Large-cell Monte Carlo renormalization group for percolation
Large-cell Monte Carlo renormalization group for percolation
复制标题
DOI：
10.1103/physrevb.21.1223
复制
发表时间：
1980-02
影响因子：
3.7
通讯作者：
P. Reynolds;H. Stanley;W. Klein
中科院分区：
文献类型：
--
作者：
P. Reynolds;H. Stanley;W. Klein
我们获得的临界参数的网站渗流问题的正方形格子上的一个高精度（可比的级数展开）通过使用Monte Carlo位置空间重整化组程序直接对占位概率。我们的方法包括使用逐渐变大的格重标度B来计算递归关系。我们发现光滑序列的临界渗流浓度pc（B）的值和缩放功率yp（B）和yh（B）。将这些序列外推到极限B→∞会得到相当精确的数值预测。此外，通过考虑其他的权函数或”规则”，也体现了基本的连通性特征的渗流，我们发现，在无限单元极限的数值结果实际上是”规则无关的。“然而，接近这一限度的实际方式确实取决于所选择的规则。我们的重整化群的结果和有限尺寸标度的外推之间的连接。此外，通常的有限尺寸尺度参数导致独立的估计pc和y p。结合大细胞的方法和有限尺寸尺度的结果，我们得到y p= 0.7385±0.0080和y h= 1.898±0.003。因此我们得到α p=− 0.708±0.030，β p= 0.138（+ 0.006，− 0.005），γ p= 2.432±0.035，δ p= 18.6±0.6，ν p= 1.354±0.015，2− η p= 1.796±0.006。发现正方形晶格的位逾渗阈值为Pc = 0.5931±0.0006。我们注意到，我们计算的ν p值与Klein等人提出的ν p= ln 3 ln（3 2）<$1.3548的结果比与den Nijs最近提出的ν p= 4 3的结果更吻合。然而，我们的结果不能完全排除后一种可能性。
We obtain the critical parameters for the site-percolation problem on the square lattice to a high degree of accuracy (comparable to that of series expansions) by using a Monte Carlo position-space renormalization-group procedure directly on the site-occupation probability. Our method involves calculating recursion relations using progressively larger lattice rescalings, b. We find smooth sequences for the value of the critical percolation concentration p c (b) and for the scaling powers y p (b) and y h (b). Extrapolating these sequences to the limit b→∞ leads to quite accurate numerical predictions. Further, by considering other weight functions or" rules" which also embody the essential connectivity feature of percolation, we find that the numerical results in the infinite-cell limit are in fact" rule independent." However, the actual fashion in which this limit is approached does depend upon the rule chosen. A connection between extrapolation of our renormalization-group results and finite-size scaling is made. Furthermore, the usual finite-size scaling arguments lead to independent estimates of p c and y p. Combining both the large-cell approach and the finite-size scaling results, we obtain y p= 0.7385±0.0080 and y h= 1.898±0.003. Thus we find α p=− 0.708±0.030, β p= 0.138 (+ 0.006,− 0.005), γ p= 2.432±0.035, δ p= 18.6±0.6, ν p= 1.354±0.015, and 2− η p= 1.796±0.006. The site-percolation threshold is found for the square lattice at p c= 0.5931±0.0006. We note that our calculated value of ν p is in considerably better agreement with the proposal of Klein et al. that ν p= ln 3 ln (3 2)≅ 1.3548, than with den Nijs' recent conjecture, which predicts ν p= 4 3. However, our results cannot entirely rule out the latter possibility.