Giant component of the soft random geometric graph

Abstract

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