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Bond percolation between k separated points on a square lattice

S. S. Manna,
Published: 26 June 2020

Abstract: 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.
Keywords: lattice / percolation / k points / size / square / function / thresholds

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