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

We discuss necessary conditions for the existence of probability distribution on particle configurations in $d$-dimensions i.e. a point process, compatible with a specified density $\rho$ and radial distribution function $g({\bf r})$. In $d=1$ we give necessary and sufficient criteria on $\rho g({\bf r})$ 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 $\rho D \leq e^{-1}$: the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for $\rho g(r)$ to specify a determinantal point process for which all $n$-particle densities, $\rho_n({\bf r}_1, ..., {\bf r}_n)$, are given explicitly as determinants. We give several examples.