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Bond percolation between k separated points on a square lattice.
Physical Review E ( IF 2.2 ) Pub Date : 2020-06-26 , DOI: 10.1103/physreve.101.062143
S S Manna 1 , Robert M Ziff 2
Affiliation  

We consider a percolation process in which k points separated by a distance proportional the 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 p¯ck defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of pck for individual samples remains broad in the limit of L. We study p¯ck for bond percolation on the square lattice and find that p¯ck are above the normal percolation threshold pc=1/2 and represent specific supercritical states. The p¯ck can be related to integrals over powers of the function P(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(p) on L×L systems that for L, p¯c1=0.51755(5), p¯c2=0.53219(5), p¯c3=0.54456(5), and p¯c4=0.55527(5). The percolation thresholds 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 L1/νk where νk=ν/(kβ+1), with β=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν1=48/41, ν2=24/23, ν3=16/17, ν4=6/7, etc. We also study three-point correlations in the system and show how for p>pc, the correlation ratio goes to 1 (no net correlation) as L, while at pc it reaches the known value of 1.022.

中文翻译:

方格上k个分离点之间的键渗透。

我们考虑一个渗滤过程,其中 ķ 点之间的距离与系统大小成正比 大号 同时连接在一起(ķ>1个),或系统中心的单个点连接到边界(ķ=1个),通过单个群集的相邻连接点。这些过程产生了新的阈值p¯Cķ 定义为 p所需连接首先出现的位置。这些阈值并不尖锐,因为pCķ 单个样本的范围仍然很广 大号。我们学习p¯Cķ 对于方格上的键渗滤,发现 p¯Cķ 高于正常渗滤阈值 pC=1个/2并代表特定的超临界状态。的p¯Cķ 可能与函数的幂积分有关 Pp等于一个点连接到无限簇的概率;我们从直接模拟和测量得到的数值Pp大号×大号 用于 大号 p¯C1个=0.517555 p¯C2=0.532195 p¯C3=0.544565p¯C4=0.555275。渗滤阈值p¯Cķ 保持不变,即使 ķ点在晶格内随机选择。我们显示有限大小的校正比例为大号-1个/νķ 哪里 νķ=ν/ķβ+1个,带有 β=5/36ν=4/3 是普通的渗流临界指数,所以 ν1个=48/41 ν2=24/23 ν3=16/17 ν4=6/7等。我们还研究了系统中的三点关联,并说明了如何 p>pC,则相关比率变为1(无净相关),因为 大号,而在 pC 它达到了 1.022
更新日期:2020-06-26
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