Scaling universalities of kth-nearest neighbor distances on closed manifolds

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dc.creatorPercus, A. G.
dc.creatorMartin, O. C.
dc.date1998-02-25
dc.date.accessioned2026-07-07T05:23:56Z
dc.date.available2026-07-07T05:23:56Z
dc.descriptionTake 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, the k-dependence and the N-dependence separate in <D_k(N)>. All kth-nearest neighbor distances thus have the same scaling law in N. Second, for a curved surface, the average \int <D_k(N)> dμover 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.
dc.description14 pages, 2 figures; submitted to Advances in Applied Mathematics
dc.identifierhttps://arxiv.org/abs/math/9802117
dc.identifierhttp://arxiv.org/abs/math/9802117
dc.identifierAdvances in Applied Mathematics 21 (1998) 424-436 (1998); published version available at web page http://www.lanl.gov/home/percus
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/76645
dc.subjectDifferential Geometry
dc.subjectGeometric Topology
dc.subject60D05 (Primary); 51H25 (Secondary)
dc.titleScaling universalities of kth-nearest neighbor distances on closed manifolds
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