On the first and second largest components in the percolated random geometric graph

Abstract

The percolated random geometric graph Gn(λ,p) has vertex set given by a Poisson Point Process in the square [0,n]2, and every pair of vertices at distance at most 1 independently forms an edge with probability p. For a fixed p, Penrose proved that there is a critical intensity λc=λc(p) for the existence of a giant component in Gn(λ,p). Our main result shows that for λ>λc, the size of the second-largest component is a.a.s. of order (logn)2. Moreover, we prove that the size of the largest component rescaled by n converges almost surely to a constant, thereby strengthening results of Penrose. We complement our study by showing a certain duality result between percolation thresholds associated to the Poisson intensity and the bond percolation of G(λ,p) (which is the infinite volume version of Gn(λ,p)). Moreover, we prove that for a large class of graphs converging in a suitable sense to G(λ,1), the corresponding critical percolation thresholds converge as well to the ones of G(λ,1).