Analytical calculation of neighborhood order probabilities for high dimensional Poissonic processes and mean field models

TL;DR

This paper derives analytical formulas for neighborhood order probabilities in high-dimensional Poisson point processes, highlighting the role of dimensionality and connecting to models like the random link and random map models.

Contribution

It provides a simplified derivation of Cox probabilities emphasizing system dimensionality and extends results to the random link and random map models.

Findings

Closed-form Cox probabilities in high dimensions

Analytical results for finite and infinite systems

Connection between spatial statistics and disordered media models

Abstract

Consider that the coordinates of $N$ points are randomly generated along the edges of a $d$-dimensional hypercube (random point problem). The probability that an arbitrary point is the $m$th nearest neighbor to its own $n$th nearest neighbor (Cox probabilities) plays an important role in spatial statistics. Also, it has been useful in the description of physical processes in disordered media. Here we propose a simpler derivation of Cox probabilities, where we stress the role played by the system dimensionality $d$. In the limit $d \to \infty$, the distances between pair of points become indenpendent (random link model) and closed analytical forms for the neighborhood probabilities are obtained both for the thermodynamic limit and finite-size system. Breaking the distance symmetry constraint drives us to the random map model, for which the Cox probabilities are obtained for two cases: whether a point is its own nearest neighbor or not.