Giant component of the soft random geometric graph

Abstract

Consider a 2-dimensional soft random geometric graph $G(\mathit{\lambda },s,\mathit{\varphi })$, obtained by placing a Poisson($\mathit{\lambda }{s}^2$) number of vertices uniformly at random in a square of side s, with edges placed between each pair $x,y$ of vertices with probability $\mathit{\varphi }(\Vert x-y\Vert )$, where $\mathit{\varphi }:{\mathbb{R}}_{+}\to [0,1]$ is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph $G(\mathit{\lambda },s,\mathit{\varphi })$ in the large-s limit with $(\mathit{\lambda },\mathit{\varphi })$ 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 ${\mathit{\lambda }}_c(\mathit{\varphi })$.