Largest nearest-neighbour link and connectivity threshold in a polytopal random sample

Abstract

Let $$X_1,X_2, \ldots $$ X 1 , X 2 , … be independent identically distributed random points in a convex polytopal domain $$A \subset \mathbb {R}^d$$ A ⊂ R d . Define the largest nearest-neighbour link$$L_n$$ L n to be the smallest r such that every point of $$\mathscr {X}_n:=\{X_1,\ldots ,X_n\}$$ X n : = { X 1 , … , X n } has another such point within distance r. We obtain a strong law of large numbers for $$L_n$$ L n in the large-n limit. A related threshold, the connectivity threshold$$M_n$$ M n , is the smallest r such that the random geometric graph $$G(\mathscr {X}_n, r)$$ G ( X n , r ) is connected (so $$L_n \le M_n$$ L n ≤ M n ). We show that as $$n \rightarrow \infty $$ n → ∞ , almost surely $$nL_n^d/\log n$$ n L n d / log n tends to a limit that depends on the geometry of A, and $$nM_n^d/\log n$$ n M n d / log n tends to the same limit. We derive these results via asymptotic lower bounds for $$L_n$$ L n and upper bounds for $$M_n$$ M n that are applicable in a larger class of metric spaces satisfying certain regularity conditions.