Monotonicity of Percolation Probability on ${\mathbb{Z}}^d$ with Long Range Connections

Consider an independent site percolation model on ${\mathbb{Z}}^d,d\ge 2$, with parameter $p\in (0,1)$, where there are only nearest neighbor bonds and long range bonds of length $k$ parallel to some coordinate axis. We show that the percolation probability is a non-decreasing function of $k$. We also generalize this result for models whose long range bonds have several lengths.