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

Lyuben Lichev, Bas Lodewijks, Dieter Mitsche and Bruno Schapira

Stochastic Processes and their Applications, 2023, vol. 164, issue C, 311-336

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).

Keywords: Random geometric graph; Second-largest component; Giant component; Continuum percolation; Bond percolation; Schramm’s locality conjecture (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.07.008

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