SCALING UNIVERSALITIES OF KTH-NEAREST NEIGHBOR DISTANCES ON CLOSED MANIFOLDS

Take N sites distributed randomly and uniformly on a smooth closed sur face. We express the expected distance [D-k(N)] from an arbitrary poin t on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius I about that point. We then find two universalities. First, for a flat surface, where A(I) = pi/(2). [D-k(N)] is separable in k and N. All kth-nearest neighbor d istances thus scale the same way in N;. Second, for a curved surface, [D-k(N)] averaged over the surface is a topological invariant at leadi ng and subleading order in a large N expansion. The 1/N scaling series then depends, up through O(1/N), only on the surface's topology and n ot on its precise shape. Pie discuss the case of higher dimensions (d > 2), and also interpret our results using Regge calculus. (C) 1998 Ac ademic Press.