Constructing, characterizing, and simulating Gaussian and higher-order point distributions

The definition and the properties of a Gaussian point distribution, in contrast to the well-known properties of a Gaussian random field are discussed. Constraints for the number density and the two-point correlation function arise. A simple method for the simulation of this so-called Gauss-Poisson point process is given and illustrated with an example. A comparison of the distribution of galaxies in the PSCz catalog with the Gauss-Poisson process underlines the importance of higher-order correlation functions for the description for the galaxy distribution. The construction of the Gauss-Poisson point process is extended to the n-point Poisson cluster process, now incorporating correlation functions up to $n\mathrm{th}$ order. Simulation methods and constraints on the correlation functions are discussed for an n-point case and detailed for a three-point case. As another approach, well suited for strongly clustered systems, the generalized halo model is discussed. The influence of substructure inside the halos on the two- and three-point correlation functions is calculated in this model.