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

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 (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.