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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.24624 |
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| _version_ | 1866913065825468416 |
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| author | Otsuka, Junpei |
| author_facet | Otsuka, Junpei |
| contents | A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate the limit distribution of the threshold radius for which the maximum degree of the graph is at least a given value that depends on $n$. In addition, given the radii $(r_n)_{n \in \mathbf{N}}$, we examine the limiting behavior of the point process formed by the vertices that achieve the maximum degree. Roughly speaking, the limiting process exhibits a compound Poisson behavior in the regime where the maximum degree remains bounded, due to local geometric dependencies, whereas it exhibits a Poisson behavior in the regime where the maximum degree diverges more slowly than $\log n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24624 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Limit distributions of the threshold radius for the maximum degree and the associated point configurations in random geometric graphs Otsuka, Junpei Probability A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate the limit distribution of the threshold radius for which the maximum degree of the graph is at least a given value that depends on $n$. In addition, given the radii $(r_n)_{n \in \mathbf{N}}$, we examine the limiting behavior of the point process formed by the vertices that achieve the maximum degree. Roughly speaking, the limiting process exhibits a compound Poisson behavior in the regime where the maximum degree remains bounded, due to local geometric dependencies, whereas it exhibits a Poisson behavior in the regime where the maximum degree diverges more slowly than $\log n$. |
| title | Limit distributions of the threshold radius for the maximum degree and the associated point configurations in random geometric graphs |
| topic | Probability |
| url | https://arxiv.org/abs/2604.24624 |