Giant component of the soft random geometric graph

Consider a 2-dimensional soft random geometric graph G(lambda, s, phi), obtained by placing a Poisson(lambda s(2)) number of vertices uniformly at random in a square of side s, with edges placed between each pair x, y of vertices with probability phi(parallel to x - y parallel to), where phi : R+ -> [0, 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G(lambda, s, phi) in the large-s limit with (lambda, phi) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where lambda equals the critical value lambda(c)(phi)