Transition energy fields in the method of correlation equations

TL;DR

This paper generalizes the method of correlation equations for Gibbs measures by introducing transition energy fields, establishing conditions for unique solutions, and proving convergence of finite-volume correlations.

Contribution

It introduces the concept of transition energy fields into the correlation equations framework, providing new conditions for existence and uniqueness of solutions.

Findings

Unique solutions for correlation functions under small transition energies

Convergence of finite-volume correlation functions to the infinite-volume limit

Extension of correlation equations method using transition energy fields

Abstract

In this paper, the well-known method of correlation equations for constructing Gibbs measures is generalized based on the concept of the transition energy field. Using the properties of transition energies, we obtain the system of correlation equations for lattice systems with finite spin space. It is shown that for a sufficiently small value of the one-point transition energies, the corresponding system of correlation functions, considered in infinite space, has a solution which is unique. Finally, the convergence of finite-volume correlation functions to the limiting correlation function is shown.