Scaling Universalities ofkth-Nearest Neighbor Distances on Closed Manifolds
✍ Scribed by Allon G Percus; Olivier C Martin
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 247 KB
- Volume
- 21
- Category
- Article
- ISSN
- 0196-8858
No coin nor oath required. For personal study only.
✦ Synopsis
Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance D k N 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 = πl 2 , D k N is separable in k and N. All kth-nearest neighbor distances thus scale the same way in N. Second, for a curved surface, D k N 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 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.