2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125205Consider 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.12 pages, 3 figures and 1 tableDisordered Systems and Neural NetworksStatistical Mechanicstext