  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.
