On the scale free laws of urban facilities

•We implement a double stochastic process model for point patterns of urban facilities.•The model with power covariance function fits best to many statistical facts.•The power law fits best to the mean-variance relationship.•The power law fits best to the distribution function H, G, and F.

We implement a double stochastic process as the mathematical model for the spatial point patterns of urban facilities. We find that the model with power covariance function can produce the best fit not only to K   function (whose derivative gives the radial distribution ρ(t)=K′(t)/2πtρ(t)=K′(t)/2πt) but also to additional facts of spatial point patterns. These facts include the mean-variance relationship of number of events in a series of expanding bins, and other statistics beyond the first two orders, such as inter-event distribution function H(t)H(t) and nearest neighbor distribution functions G(t)G(t) and F(t)F(t).