Some peculiar boundary phenomena for extremes of rth nearest neighbor links

Let D$_{n,r}$ denote the largest rth nearest neighbor link for n points drawn independently and uniformly from the unit d-cube C$_d$. We show that according as r < d or r>d, the limiting behavior of D$_{n,r}$, as n → ∞, is determined by the two-dimensional ‘faces’ respectively one-dimensional ‘edges’ of the boundary of C$_d$. If d = r, a ‘balance’ between faces and edges occurs. In case of a d-dimensional sphere (instead of a cube) the boundary dominates the asymptotic behavior of D$_{n,r}$ if d ⩾ 3 or if d = 2, r ⩾ 3.