Bond percolation between k separated points on a square lattice
Abstract: We consider a percolation process in which points separated by a distance proportional the system size simultaneously connect together (), or a single point at the center of a system connects to the boundary (), through adjacent connected points of a single cluster. These processes yield new thresholds defined as the average value of at which the desired connections first occur. These thresholds not sharp, as the distribution of values of for individual samples remains broad in the limit of . We study for bond percolation on the square lattice and find that are above the normal percolation threshold and represent specific supercritical states. The can be related to integrals over powers of the function equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of on systems that for , and . The percolation thresholds remain the same, even when the points are randomly selected within the lattice. We show that the finite-size corrections scale as where , with and being the ordinary percolation critical exponents, so that , etc. We also study three-point correlations in the system and show how for , the correlation ratio goes to 1 (no net correlation) as , while at it reaches the known value of .
Keywords: lattice / percolation / k points / size / square / function / thresholds
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