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

Consider an independent site percolation model with parameter p ∈ ( 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 2 d ) when k goes to infinity, the percolation threshold for ordinary (nearest neighbor) percolation on Z 2 d . We also generalize this result for models whose long range bonds have several lengths.