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

Consider an independent site percolation model with parameter p[set membership, variant](0,1) on , 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 when k goes to infinity, the percolation threshold for ordinary (nearest neighbor) percolation on . We also generalize this result for models whose long range bonds have several lengths.