On $k$-clusters of high-intensity random geometric graphs
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916680555298816 |
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| author | Penrose, Mathew D. Yang, Xiaochuan |
| author_facet | Penrose, Mathew D. Yang, Xiaochuan |
| contents | Let $k,d $ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$-dimensional Poisson Boolean model with balls of fixed radius is of order $k$, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a random geometric graph on $n$ uniformly distributed vertices in a smoothly bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that the expected degree grows slowly as $n$ becomes large (the so-called mildly dense limiting regime). We also show that the variance of $S_{n,k}$ is asymptotic to its mean, and prove Poisson and normal approximation results for $S_{n,k}$ in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points).
We also give similar results in the so-called mildly sparse limiting regime where $r(n)$ is chosen so the expected degree decays slowly to zero as $n $ becomes large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_14758 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On $k$-clusters of high-intensity random geometric graphs Penrose, Mathew D. Yang, Xiaochuan Probability 05C80, 60K35 Let $k,d $ be positive integers. We determine a sequence of constants that are asymptotic to the probability that the cluster at the origin in a $d$-dimensional Poisson Boolean model with balls of fixed radius is of order $k$, as the intensity becomes large. Using this, we determine the asymptotics of the mean of the number of components of order $k$, denoted $S_{n,k}$ in a random geometric graph on $n$ uniformly distributed vertices in a smoothly bounded compact region of $R^d$, with distance parameter $r(n)$ chosen so that the expected degree grows slowly as $n$ becomes large (the so-called mildly dense limiting regime). We also show that the variance of $S_{n,k}$ is asymptotic to its mean, and prove Poisson and normal approximation results for $S_{n,k}$ in this limiting regime. We provide analogous results for the corresponding Poisson process (i.e. with a Poisson number of points). We also give similar results in the so-called mildly sparse limiting regime where $r(n)$ is chosen so the expected degree decays slowly to zero as $n $ becomes large. |
| title | On $k$-clusters of high-intensity random geometric graphs |
| topic | Probability 05C80, 60K35 |
| url | https://arxiv.org/abs/2209.14758 |