Consider a 2-dimensional soft random geometric graph , obtained by placing a Poisson() number of vertices uniformly at random in a square of side s, with edges placed between each pair of vertices with probability , where is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph in the large-s limit with 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 .
Supported by EPSRC grant EP/T028653/1.
"Giant component of the soft random geometric graph." Electron. Commun. Probab. 27 1 - 10, 2022. https://doi.org/10.1214/22-ECP491