An Inverse Problem for Gibbs Fields with Hard Core Potential

It is well known that for a regular stable potential of pair interaction and a small value of activity one can define the corresponding Gibbs field (a measure on the space of configurations of points in ${\mathbb{R}}^d$). In this paper we consider a converse problem. Namely, we show that for a sufficiently small constant ${\bar{\rho }}_1$ and a sufficiently small function ${\bar{\rho }}_2(x)$, $x\in {\mathbb{R}}^d$, that is equal to zero in a neighborhood of the origin, there exist a hard core pair potential, and a value of activity, such that ${\bar{\rho }}_1$ is the density and ${\bar{\rho }}_2$ is the pair correlation function of the corresponding Gibbs field.