The Existence of Pair Potential Corresponding to Specified Density and Pair Correlation

Given a potential of pair interaction and a value of activity, one can consider the Gibbs distribution in a finite domain $\Lambda \subset \mathbb{Z}^d$. It is well known that for small values of activity there exist the infinite volume ($\Lambda \to \mathbb{Z}^d$) limiting Gibbs distribution and the infinite volume correlation functions. In this paper we consider the converse problem - we show that given $\rho_1$ and $\rho_2(x)$, where $\rho_1$ is a constant and $\rho_2(x)$ is a function on $\mathbb{Z}^d$, which are sufficiently small, there exist a pair potential and a value of activity, for which $\rho_1$ is the density and $\rho_2(x)$ is the pair correlation function.