On the Construction of Particle Distributions with Specified Single and Pair Densities

We discuss necessary conditions for the existence of a probability distribution on particle configurations in d -dimensions, i.e., a point process, compatible with a specified density ρ and radial distribution function g ( r ). In d = 1 we give necessary and sufficient criteria on ρ g ( r ) for the existence of such a point process of renewal (Markov) type. We prove that these conditions are satisfied for the case g ( r ) = 0, r < D and g ( r ) = 1, r > D, if and only if ρ D ≤ e -1: the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for ρ g ( r ) to specify a determinantal point process for which all n -particle densities,ρ n ( r 1,..., r n ), are given explicitly as determinants. We give several examples.