Spread-out percolation in ℝ\u3csup\u3ed\u3c/sup\u3e

Fix d ≥ 2, and let X be either ℤd or the points of a Poisson process in ℝd of intensity 1. Given parameters r and p, join each pair of points of X within distance r independently with probability p. This is the simplest case of a spread-out percolation model studied by Penrose [Ann Appl Probab 3 (1993) 253-276], who showed that, as r -→ ∞, the average degree of the corresponding random graph at the percolation threshold tends to l, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of [3] on inhomogeneous random graphs. © 2007 Wiley Periodicals, Inc