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Title: | Largest nearest-neighbour link and connectivity threshold in a polytopal random sample |
Authors: | Penrose, MD Yang, X |
Keywords: | probability;math.PR;60D05;60F15;05C80 |
Issue Date: | 6-Jan-2023 |
Publisher: | Cornell University |
Citation: | Penrose, M.D. and Yang, X. (2023) 'Largest nearest-neighbour link and connectivity threshold in a polytopal random sample', arXiv:2301.02506v1 [math.PR], pp. 1 - 26. doi: 10.48550/arXiv.2301.02506. |
Abstract: | Copyright © 2023 The Authors. Let $X_1,X_2, \ldots $ be independent identically distributed random points in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest nearest neighbour link $L_n$ to be the smallest $r$ such that every point of $\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$. We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A related threshold, the connectivity threshold $M_n$, is the smallest $r$ such that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit. |
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URI: | https://bura.brunel.ac.uk/handle/2438/25870 |
DOI: | https://doi.org/10.48550/arXiv.2301.02506 |
Other Identifiers: | ORCID iD: Xiaochuan Yang https://orcid.org/0000-0003-2435-4615 https://arxiv.org/abs/2301.02506v1 |
Appears in Collections: | Dept of Mathematics Research Papers |
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Preprint.pdf | Copyright © 2023 The Authors. Submitted to arXiv under a Creative Commons 4.0 International (CC BY 4.0) Attribution License (https://creativecommons.org/licenses/by/4.0/) | 269.72 kB | Adobe PDF | View/Open |
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