Line-of-Sight Percolation

Given omega >= 1, let Z((omega))(2) be the graph with vertex Set Z(2) in which two vertices are joined if they agree in one coordinate and differ by at most omega in the other. (Thus Z((1))(2) is precisely Z(2).) Let p(c)(omega) be the critical probability for site percolation on Z((omega))(2) Extending recent results of Frieze, Kleinberg, Ravi and Debany, we show that lim(omega ->infinity) omega p(c)(omega) = log(3/2). We also prove analogues of this result for the n-by-n grid and in higher dimensions, the latter involving interesting connections to Gilbert's continuum percolation model. To prove our results, we explore the component of the origin in a certain non-standard way, and show that this exploration is well approximated by a certain branching random walk.