O. Costin and J.L. Lebowitz
On the Construction of Particle Distributions with Specified Single 
and Pair Densities
(31K, Latex)

ABSTRACT.  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.