Boundary Conditions and Numbers of Clusters in Percolation Models

Abstract

A histogram Monte Carlo simulation method is used to calculate numbers of percolating and nonpercolating clusters on finite planar lattices with different boundary conditions. We have found that the average number of percolating clusters, C, increases linearly with the aspect ratio, R, of the lattice for large R, i.e. in this case C = aR with a constant a. We have also found that a is independent of the boundary conditions. For the periodic boundary conditions in both horizontal and vertical directions, we have confirmed Ziff et. al. result that the number of clusters per lattice site, n, for percolation on two dimensional lattices with linear dimensions L may be written as n = n c + b/N, where n c is n in the limit L → ∞, b is a constant and N is the number of lattice sites. ZifF et. al. found that b is universal and presented an argument that b is the number of percolating clusters so that the universality of b may be related to the universality of C found by Hu and Lin. We have found that for large R, b = b c R, but b c ^ a.KeywordsAspect RatioVertical DirectionLattice SiteLinear DimensionLinear SlopeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.