论文标题
$ k $之间的债券渗透在平方晶格上分离点
Bond percolation between $k$ separated points on a square lattice
论文作者
论文摘要
我们考虑了一个渗透过程，其中$ k $点以与系统尺寸$ l $的距离相距（$ k> 1 $），或系统中心的一个点连接到边界（$ k = 1 $），通过单个群集的相邻连接点连接。这些过程产生了新的阈值$ \ operline p_ {ck} $定义为首先发生所需连接的$ p $的平均值。这些阈值并不明确，因为单个样本的$ p_ {ck} $的值分布在$ l \ to \ ftty $的极限中仍然很大。我们研究了方格上的债券渗透的$ \ OVILLINE P_ {CK} $，发现$ \ Overline P_ {CK} $超过了正常的渗透阈值$ P_C = 1/2 $并表示特定的超临界状态。 $ \ OVILLINE P_ {CK} $可以与功能$ p_ \ infty（p）$相当于概率的函数的积分相关，等于概率是一个点连接到Infinite cluster的概率；我们从直接仿真和$ p_ \ infty（p）$的测量中找到了$ l \ times l $ $系统的测量，这些系统对于$ l \ to \ to \ infty $，$ \ overline p_ {c1} = 0.51755（5）$ 0.54456（5）$，$ \ overline p_ {c4} = 0.55527（5）。$ percolation阈值$ \ overline p_ {ck} $保持相同，即使$ k $点在晶格中随机选择。 We show that the finite-size corrections scale as $L^{-1/ν_k}$ where $ν_k = ν/(k β+1)$, with $β=5/36$ and $ν=4/3$ being the ordinary percolation critical exponents, so that $ν_1= 48/41$, $ν_2 = 24/23$, $ν_3 = 16/17$, $ν_4 = 6/7 $等。我们还研究了系统中的三点相关性，并显示$ p> p_c $的相关比如何为1（无净相关）为$ l \ to \ forty $，而在$ p_c $中，它达到已知的值为1.022。
We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through adjacent connected points of a single cluster. These processes yield new thresholds $\overline p_{ck}$ defined as the average value of $p$ at which the desired connections first occur. These thresholds are not sharp as the distribution of values of $p_{ck}$ for individual samples remains broad in the limit of $L \to \infty$. We study $\overline p_{ck}$ for bond percolation on the square lattice, and find that $\overline p_{ck}$ are above the normal percolation threshold $p_c = 1/2$ and represent specific supercritical states. The $\overline p_{ck}$ can be related to integrals over powers of the function $P_\infty(p)$ equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of $P_\infty(p)$ on $L\times L$ systems that, for $L \to \infty$, $\overline p_{c1} = 0.51755(5)$, $\overline p_{c2} = 0.53219(5)$, $\overline p_{c3} = 0.54456(5)$, and $\overline p_{c4} = 0.55527(5).$ The percolation thresholds $\overline p_{ck}$ remain the same, even when the $k$ points are randomly selected within the lattice. We show that the finite-size corrections scale as $L^{-1/ν_k}$ where $ν_k = ν/(k β+1)$, with $β=5/36$ and $ν=4/3$ being the ordinary percolation critical exponents, so that $ν_1= 48/41$, $ν_2 = 24/23$, $ν_3 = 16/17$, $ν_4 = 6/7$, etc. We also study three-point correlations in the system, and show how for $p>p_c$, the correlation ratio goes to 1 (no net correlation) as $L \to \infty$, while at $p_c$ it reaches the known value of 1.022.
