Abstract
Consider a 2-dimensional soft random geometric graph G(λ,s,ϕ), obtained by placing a Poisson(λs2) 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(‖x−y‖), 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 λc(ϕ).
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