Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913569807794176 |
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| author | Sambale, Holger Thäle, Christoph Trauthwein, Tara |
| author_facet | Sambale, Holger Thäle, Christoph Trauthwein, Tara |
| contents | Consider a stationary Poisson process $η$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $η$ as follows. First, each point $x\inη$ is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until $x$ is contained in the convex hull of the points already connected to $x$. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00748 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space Sambale, Holger Thäle, Christoph Trauthwein, Tara Probability 60D05, 60F05, 60G55 Consider a stationary Poisson process $η$ in the $d$-dimensional Euclidean or hyperbolic space and construct a random graph with vertex set $η$ as follows. First, each point $x\inη$ is connected by an edge to its nearest neighbour, then to its second nearest neighbour and so on, until $x$ is contained in the convex hull of the points already connected to $x$. The resulting random graph is the so-called nearest neighbour embracing graph. The main result of this paper is a quantitative description of the Gaussian fluctuations of geometric functionals associated with the nearest neighbour embracing graph. More precisely, the total edge length, more general length-power functionals and the number of vertices with given outdegree are considered. |
| title | Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space |
| topic | Probability 60D05, 60F05, 60G55 |
| url | https://arxiv.org/abs/2411.00748 |