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DC Field | Value | Language |
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dc.contributor.author | Penrose, MD | - |
dc.contributor.author | Yang, X | - |
dc.date.accessioned | 2023-01-24T17:31:53Z | - |
dc.date.available | 2023-01-24T17:31:53Z | - |
dc.date.issued | 2023-01-06 | - |
dc.identifier | ORCID iD: Xiaochuan Yang https://orcid.org/0000-0003-2435-4615 | - |
dc.identifier | https://arxiv.org/abs/2301.02506v1 | - |
dc.identifier.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. | en_US |
dc.identifier.uri | https://bura.brunel.ac.uk/handle/2438/25870 | - |
dc.description | Material is not peer-reviewed by arXiv - the contents of arXiv submissions are wholly the responsibility of the submitter and are presented “as is” without any warranty or guarantee. By hosting works and other materials on this site, arXiv, Cornell University, and their agents do not in any way convey implied approval of the assumptions, methods, results, or conclusions of the work. | en_US |
dc.description.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. | en_US |
dc.description.sponsorship | EPSRC grant EP/T028653/1. | en_US |
dc.format.extent | 1 - 26 | - |
dc.format.medium | Electronic | - |
dc.language.iso | en_US | en_US |
dc.publisher | Cornell University | en_US |
dc.rights | 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/). You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially. This license is acceptable for Free Cultural Works. The licensor cannot revoke these freedoms as long as you follow the license terms. | - |
dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | - |
dc.subject | probability | en_US |
dc.subject | math.PR | en_US |
dc.subject | 60D05 | en_US |
dc.subject | 60F15 | en_US |
dc.subject | 05C80 | en_US |
dc.title | Largest nearest-neighbour link and connectivity threshold in a polytopal random sample | en_US |
dc.type | Article | en_US |
dc.identifier.doi | https://doi.org/10.48550/arXiv.2301.02506 | - |
pubs.notes | 26 pages | - |
dc.identifier.eissn | 2331-8422 | - |
dc.rights.holder | The Authors | - |
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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