Scaling universalities of kth-nearest neighbor distances on closed manifolds

Details Scaling universalities of kth-nearest neighbor distances on closed manifolds Scaling universalities of kth-nearest neighbor distances on closed manifolds

1998-02-25

arxiv.org/abs/math/9802117v1

Advances in Applied Mathematics 21 (1998) 424-436 (1998); published version available at web page http://www.lanl.gov/home/per

Mathematics

Differential Geometry

14 pages, 2 figures; submitted to Advances in Applied Mathematics

Scientific paper

Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius l about that point. We then find two universalities. First, for a flat surface, where A(l)=\pi l^2, the k-dependence and the N-dependence separate in . All kth-nearest neighbor distances thus have the same scaling law in N. Second, for a curved surface, the average \int d\mu 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 O(1/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.

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