Critical Point and Percolation Probability in a Long Range Site   Percolation Model on $\Z^d$
  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. 
Bernardo N. B. de Lima, R\'emy Sanchis, Roger W. C. Silva
math.PR