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 Λ ⊂ Z. It is well known that for small values of activity there exist the infinite volume (Λ → Z) limiting Gibbs distribution and the infinite volume correlation functions. In this paper we consider the converse problem we show that given ρ1 and ρ2(x), where ρ1 is a constant and ρ2(x) is a function on Z, which are sufficiently small, there exist a pair potential and a value of activity, for which ρ1 is the density and ρ2(x) is the pair correlation function.