Regular Article: Scaling Universalities ofkth-Nearest Neighbor Distances on Closed Manifolds

  • Authors:

  • Allon G Percus;Olivier C Martin

  • Affiliations:

  • CIC-3 and Center for Nonlinear Studies, MS-B258, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545;Division de Physique Théorique, Institut de Physique Nucléaire, Université Paris-Sud, F-91406, Orsay Cedex, France

  • Venue:
  • Advances in Applied Mathematics
  • Year:
  • 1998

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Abstract

TakeNsites distributed randomly and uniformly on a smooth closed surface. We express the expected distance 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)=@pl^2, is separable inkandN. Allkth-nearest neighbor distances thus scale the same way inN. Second, for a curved surface, 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.