Abstract
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 \((\Lambda \rightarrow \mathbb{Z}^{d})\) 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 \(\mathbb{Z}^{d}\) , 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.
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Koralov, L. Existence of Pair Potential Corresponding to Specified Density and Pair Correlation. Lett Math Phys 71, 135–148 (2005). https://doi.org/10.1007/s11005-005-0343-9
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DOI: https://doi.org/10.1007/s11005-005-0343-9