Giant component of the soft random geometric graph

Consider a 2-dimensional soft random geometric graph G ( λ, s, φ ), obtained by placing a Poisson( λs 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 φ ( (cid:107) x − y (cid:107) ), where φ : R + → [0 , 1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G ( λ, s, φ ) in the large- s limit with ( λ, φ ) 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 λ equals the critical value λ c ( φ ).