Critical point and percolation probability in a long range site percolation model on Zd

Consider an independent site percolation model with parameter p is an element of (0, 1) on Z(d). d >= 2, where there are only nearest neighbor bonds and long range bonds of length k parallel to each coordinate axis. We show that the percolation threshold of such a model converges to p(c) (Z(2d)) when k goes to infinity, the percolation threshold for ordinary (nearest neighbor) percolation on Z(2d). We also generalize this result for models whose long range bonds have several lengths. (C) 2011 Elsevier B.V. All rights reserved.