A Strong Law for the Longest Edge of the Minimal Spanning Tree

Suppose X 1 , X 2 , X are independent random points in Rd, d > 2, with common density f, having connected compact support Ω with smooth boundary ∂Ω, with f|Ω 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 θ denote the volume of the unit ball. Then as n → ∞, nθM d a /log n → max((min f) -1 , 2(1 - 1/d)(min f) -1 ), a.s.