On the first and second largest components in the percolated Random Geometric Graph

Abstract

The percolated random geometric graph $G_n(\lambda, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, 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 $\lambda_c = \lambda_c(p)$ for the existence of a giant component in $G_n(\lambda, p)$. Our main result shows that for $\lambda > \lambda_c$, the size of the second-largest component is a.a.s. of order $(\log n)^2$. 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 $G(\lambda, p)$ (which is the infinite volume version of $G_n(\lambda,p)$). Moreover, we prove that for a large class of graphs converging in a suitable sense to $G(\lambda, 1)$, the corresponding critical percolation thresholds converge as well to the ones of $G(\lambda,1)$.