Eur. Phys. J. B (2014) 87: 87
https://doi.org/10.1140/epjb/e2014-50003-y

Regular Article

Distance statistics in random media

High dimension and/or high neighborhood order cases

¹ Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP), Universidade de São Paulo (USP), Avenida Bandeirantes, 3900, 14040-901 Ribeirão Preto, São Paulo, Brazil
² National Institute of Science and Technology in Complex Systems (LNCT-SC), Universidade de São Paulo (USP), Avenida Bandeirantes, 3900, 14040-901 Ribeirão Preto, São Paulo, Brazil

^a e-mail: c.roberto.fg@gmail.com

Received: 3 January 2014
Published online: 9 April 2014

Abstract

Consider an unlimited homogeneous medium disturbed by points generated via Poisson process. The neighborhood of a point plays an important role in spatial statistics problems. Here, we obtain analytically the distance statistics to kth nearest neighbor in a d-dimensional media. Next, we focus our attention in high dimensionality and high neighborhood order limits. High dimensionality makes distance distribution behavior as a delta sequence, with mean value equal to Cerf’s conjecture. Distance statistics in high neighborhood order converges to a Gaussian distribution. The general distance statistics can be applied to detect departures from Poissonian point distribution hypotheses as proposed by Thompson and generalized here.