Scaling universalities of kth-nearest neighbor distances on closed manifolds

A. G. Percus 1, O. C. Martin 2

Advances in Applied Mathematics 21 (1998) 424-436

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.

  • 1. CIC-3 and Center for Nonlinear Studies,
    Los Alamos National Laboratory
  • 2. Division de Physique Théorique, IPN,
    Université Paris XI - Paris Sud