Abstract
Suppose $X_1, X_2, X_3,\\ldots$ are independent random points in $\\mathbf{R}^d,d\\geq 2$, with common density $f$, having connected compact support $\\Omega$ with smooth boundary $\\partial\\Omega$, with $f|_{\\Omega}$ continuous. Let $M_{n}$ denote the smallest $r$ such that the union of balls of diameter $r$ centered at the first $n$ points is connected. Let $\\theta$ denote the volume of the unit ball. Then as $n\\to\\infty$, $$n\\theta M^d_n/\\log n \\to \\max\\Big(\\big(\\min\\limits_{\\Omega}f\\big)^{-1},2(1 - 1/d)\\big(\\min\\limits_{\\partial\\Omega}f\\big)^{-1}\\Big),\\quad\\text{a.s.}$$
Keywords
MathematicsOmegaCombinatoricsUnit sphereBall (mathematics)Unit (ring theory)Boundary (topology)Spanning treeGeometryMathematical analysisPhysics
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Publication Info
- Year
- 1999
- Type
- article
- Volume
- 27
- Issue
- 1
- Citations
- 117
- Access
- Closed
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Cite This
Mathew D. Penrose
(1999).
A Strong Law for the Longest Edge of the Minimal Spanning Tree.
The Annals of Probability
, 27
(1)
.
https://doi.org/10.1214/aop/1022677261
Identifiers
- DOI
- 10.1214/aop/1022677261