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

TL;DR

This paper investigates the size of the largest and second-largest components in a percolated random geometric graph, establishing asymptotic behaviors and a duality between percolation thresholds, thereby extending and strengthening prior results.

Contribution

It provides new asymptotic results for component sizes in percolated random geometric graphs and explores duality and convergence of percolation thresholds, advancing understanding beyond previous work.

Findings

Second-largest component size is of order (log n)^2 above the critical threshold.

Largest component size scaled by n converges almost surely to a constant.

Established duality between percolation thresholds of different models.

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 $G(\lambda,1)$.