An efficient algorithm to generate large random uncorrelated Euclidean distances: the random link model

A disordered medium is often constructed by $N$ points independently and identically distributed in a $d$-dimensional hyperspace. Characteristics related to the statistics of this system is known as the random point problem. As $d \to \infty$, the distances between two points become independent random variables, leading to its mean field description: the random link model. While the numerical treatment of large random point problems pose no major difficulty, the same is not true for large random link systems due to Euclidean restrictions. Exploring the deterministic nature of the congruential pseudo-random number generators, we present techniques which allow the consideration of models with memory consumption of order O(N), instead of $O(N^2)$ in a naive implementation but with the same time dependence $O(N^2)$.Comment: 8 pages, 2 figures and 1 tabl

Distance statistics in random media: high dimension and/or high neighborhood order cases

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 $k$th 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.Comment: 5 pages and 2 figure