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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2205.10923 |
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| _version_ | 1866908547380412416 |
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| author | Lichev, Lyuben Lodewijks, Bas Mitsche, Dieter Schapira, Bruno |
| author_facet | Lichev, Lyuben Lodewijks, Bas Mitsche, Dieter Schapira, Bruno |
| contents | The percolated random geometric graph $G_n(λ, p)$ 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 $λ_c = λ_c(p)$ for the existence of a giant component in $G_n(λ, p)$. Our main result shows that for $λ> λ_c$, 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 $G(λ, p)$ (which is the infinite volume version of $G_n(λ,p)$). 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 $G(λ,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10923 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the first and second largest components in the percolated Random Geometric Graph Lichev, Lyuben Lodewijks, Bas Mitsche, Dieter Schapira, Bruno Probability The percolated random geometric graph $G_n(λ, p)$ 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 $λ_c = λ_c(p)$ for the existence of a giant component in $G_n(λ, p)$. Our main result shows that for $λ> λ_c$, 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 $G(λ, p)$ (which is the infinite volume version of $G_n(λ,p)$). 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 $G(λ,1)$. |
| title | On the first and second largest components in the percolated Random Geometric Graph |
| topic | Probability |
| url | https://arxiv.org/abs/2205.10923 |