论文标题
关键指数和四个维度和五个维度的普遍过量渗透率
Critical exponents and universal excess cluster number of percolation in four and five dimensions
论文作者
论文摘要
我们通过蒙特卡洛模拟研究了周期性的四维（4D）和五维（5D）高管的临界键渗透。通过将被占用的债券分类为分支，交界和非桥梁，我们使用广度优先搜索算法构造了整个，无叶子和无桥的簇。从这些簇的几何特性中，我们确定了一组四个关键指数，包括热指数$ y _ {\ rm t} \ equiv 1/equiv 1/equiv 1/e equiv 1/en equiv 1/e equiv $ d _ {\ rm f} $，backone $ d _ {\ rm d _ {我们还获得了多余集群数量$ b $的估计，这是与簇总数的有限尺寸缩放相关的通用数量。结果为$ y _ {\ rm t} = 1.461（5）$，$ d _ {\ rm f} = 3.044 \，6（7）$，$ d _ {\ rm b} = 1.984 \，4（4（11）$，$ d _ {$ d _ {\ rm min} $ b $ b = 1.604 = 1.604 = 1.604 = 1.604， 4d;和$ y _ {\ rm t} = 1.743（10）$，$ d _ {\ rm f} = 3.526 \，0（14）$，$ d _ {\ rm b} = 2.022 \，6（27）$ 5d。关键指数的值与现有估计值兼容或改进，并且以前没有报告过多的集群数量$ b $的估计值。连同其他空间尺寸$ d $中的现有值一起，获得了关键指数的$ D $依赖性行为，并且在$ d \ d \ 5 $附近观察到了$ d _ {\ rm b} $的本地最大值。有人建议，正如预期的那样，随着$ d $的增加，关键的渗滤簇变得越来越多。
We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the leaf-free and the bridge-free clusters using the breadth-first-search algorithm. From the geometric properties of these clusters, we determine a set of four critical exponents, including the thermal exponent $y_{\rm t} \equiv 1/ν$, the fractal dimension $d_{\rm f}$, the backbone exponent $d_{\rm B}$ and the shortest-path exponent $d_{\rm min}$. We also obtain an estimate of the excess cluster number $b$ which is a universal quantity related to the finite-size scaling of the total number of clusters. The results are $y_{\rm t} = 1.461(5)$, $d_{\rm f} = 3.044 \, 6(7)$, $d_{\rm B} = 1.984\,4(11)$, $d_{\rm min} = 1.604 \, 2(5)$, $b = 0.62(1)$ for 4D; and $y_{\rm t} = 1.743(10)$, $d_{\rm f} = 3.526\,0(14)$, $d_{\rm B} = 2.022\,6(27)$, $d_{\rm min} = 1.813\, 7(16)$, $b = 0.62(2)$ for 5D. The values of the critical exponents are compatible with or improving over the existing estimates, and those of the excess cluster number $b$ have not been reported before. Together with the existing values in other spatial dimensions $d$, the $d$-dependent behavior of the critical exponents is obtained, and a local maximum of $d_{\rm B}$ is observed near $d \approx 5$. It is suggested that, as expected, critical percolation clusters become more and more dendritic as $d$ increases.
