Scaling Universalities ofkth-Nearest Neighbor Distances on Closed Manifolds

TakeNsites distributed randomly and uniformly on a smooth closed surface. We express the expected distance ?Dk(N)? from an arbitrary point on the surface to itskth-nearest neighboring site, in terms of the functionA(l) giving the area of a disc of radiuslabout that point. We then find two universalities. First, for a flat surface, whereA(l)=?l2, ?Dk(N)? is separable inkandN. Allkth-nearest neighbor distances thus scale the same way inN. Second, for a curved surface, ?Dk(N)? averaged over the surface is a topological invariant at leading and subleading order in a largeNexpansion. The 1/Nscaling series then depends, up throughO(1/N), only on the surface's topology and not on its precise shape. We discuss the case of higher dimensions (d2), and also interpret our results using Regge calculus.