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Abstract
Let X _{1}, X _{2}, … be independent identically distributed random points in a convex polytopal domain A⊂ R ^{d} . Define the largest nearestneighbour link L _{n} to be the smallest r such that every point of X _{n}: = { X _{1}, … , X _{n}} has another such point within distance r. We obtain a strong law of large numbers for L _{n} in the largen limit. A related threshold, the connectivity threshold M _{n} , is the smallest r such that the random geometric graph G(X _{n}, r) is connected (so L _{n}≤ M _{n}). We show that as n→ ∞ , almost surely nLnd/logn tends to a limit that depends on the geometry of A, and nMnd/logn tends to the same limit. We derive these results via asymptotic lower bounds for L _{n} and upper bounds for M _{n} that are applicable in a larger class of metric spaces satisfying certain regularity conditions.
Original language  English 

Journal  Journal of Applied and Computational Topology 
Early online date  16 Dec 2023 
DOIs  
Publication status  Published  16 Dec 2023 
Bibliographical note
Data availability The code used to generate Fig. 2, as well as the seeds used for the samples shown, isavailable at https://github.com/frankiehiggs/connectivityinpolytopes.
Funding
This research was funded, in part, by EPSRC Grant EP/T028653/1. A CC BY or equivalent licence is applied to the AAM arising from this submission, in accordance with the grant’s open access conditions.
Funders  Funder number 

Engineering and Physical Sciences Research Council  EP/T028653/1 
Keywords
 Random geometric graph
 Stochastic geometry
 Connectivity
 Isolated points
ASJC Scopus subject areas
 General Mathematics
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Coverage and connectivity in stochastic geometry
Penrose, M. (PI)
Engineering and Physical Sciences Research Council
15/12/20 → 15/03/25
Project: Research council