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 $ϕ(\|x-y\|)$, where $ϕ: {\bf R}_+ \to [0,1]$ 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 $λ_c(ϕ)$.

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