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

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 mth nearest neighbor to its own nth 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 → ∞, 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. PACS numbers: 05.90.+m, 02.50.Ey