  Consider an independent site percolation model with parameter $p \in (0,1)$
on $\Z^d,\ d\geq 2$ 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 model converges to $p_c(\Z^{2d})$ when $k$ goes
to infinity, the percolation threshold for ordinary (nearest neighbour)
percolation on $\Z^{2d}$. We also generalize this result for models whose long
range bonds have several lengths.

Probability