Scaling Universalities of kth-Nearest Neighbor Distances on Closed Manifolds
نویسندگان
چکیده
Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance DkN from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function Al giving the area of a disc of radius l about that point. We then find two universalities. First, for a flat surface, where Al = πl2, DkN is separable in k and N . All kth-nearest neighbor distances thus scale the same way in N . Second, for a curved surface, DkN averaged over the surface is a topological invariant at leading and subleading order in a large N expansion. The 1/N scaling series then depends, up through O1/N, only on the surface’s topology and not on its precise shape. We discuss the case of higher dimensions (d > 2), and also interpret our results using Regge calculus. © 1998 Academic Press
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