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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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Other Identifiers: ORCID iD: Xiaochuan Yang
Appears in Collections:Dept of Mathematics Research Papers

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