An Inverse Problem for Gibbs Fields with Hard Core Potential

TL;DR

This paper addresses an inverse problem in Gibbs fields, demonstrating that under certain conditions, specific density and correlation functions can be realized by a Gibbs measure with a hard core potential.

Contribution

It establishes the existence of a hard core pair potential and activity level that produce given small density and correlation functions in Gibbs fields.

Findings

Existence of a hard core potential for specified functions

Construction of Gibbs measures with prescribed properties

Conditions for small activity and correlation functions

Abstract

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 $\overline{\rho}_1$ and a sufficiently small function $\overline{\rho}_2(x)$, $x \in \mathbb{R}^d$, 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 $\overline{\rho}_1$ is the density and $\overline{\rho}_2$ is the pair correlation function of the corresponding Gibbs field.