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

Consider an independent site percolation model with parameter p∈(0,1)p∈(0,1) on Zd,d≥2, where there are only nearest neighbor bonds and long range bonds of length kk parallel to each coordinate axis. We show that the percolation threshold of such a model converges to pc(Z2d)pc(Z2d) when kk goes to infinity, the percolation threshold for ordinary (nearest neighbor) percolation on Z2dZ2d. We also generalize this result for models whose long range bonds have several lengths.

► Site Bernoulli Percolation on the d-hypercubic lattice with extra bonds with length kk.
► As kk goes to infinity the pcpc tends to the pcpc of the 2d-hypercubic.
► It is conjectured if the critical point is monotone in kk.