Random dilutions, generating functions, and the void probability distribution function

The generating function for counts in randomly placed cells is derived using the characteristic functional. The distribution of voids in a randomly diluted subset of a given point distribution is shown to contain information about all the other count probabilities. Thus random dilutions may be used to derive the entire distribution function from the void distribution. A generalization of the random dilution selection process that enhances the effects of the more massive objects on the statistics is considered. The resulting statistics are related to cluster-cluster type correlations. A relation between this selection process and the scaling properties of the distribution function of a system with scale-invariant hierarchical correlation functions is considered. Superposition of point distributions, random and homologous subsamples of point distributions, and Poisson cluster models in particular, are also discussed. Poisson cluster models are studied here because, if the peaks model of biased galaxy formation is correct, then Poisson cluster models provide a natural first approximate description of the observed galaxy distribution.