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

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 ($\Lambda \to {\mathbb{Z}}^d$) 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.

References in corpus (1)

• Determinantal random point fields

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