Showing papers by "Mathew D. Penrose published in 2023"
06 Jan 2023
TL;DR: In this article , a strong law of large numbers for L n in the large-n limit was obtained for identically distributed random points in a convex polytopal domain A ⊂ R d , where the largest nearest neighbor link L n was defined to be the smallest r such that every point of X n : = { X 1 , X 2 ,..., X n } has another such point within distance r .
Abstract: Let X 1 , X 2 ,... be independent identically distributed random points in a convex polytopal domain A ⊂ R d . Define the largest nearest neighbour 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 large- n 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. We show that as n → ∞ , almost surely nL dn / log n tends to a limit that depends on the geometry of A , and nM dn / log n tends to the same limit.