Transition energy fields in the method of correlation equations
Linda A. Khachatryan, Boris S. Nahapetian
TL;DR
The paper extends the method of correlation equations for Gibbs measures to lattice systems with finite spin spaces by formulating a transition energy field framework. It defines finite-volume correlation functions $\rho_\Lambda$ from a one-point transition energy field $\Delta_1$, derives a nonlinear correlation equation involving $\mathscr{K}$ and a boundary-term operator, and proves contraction under a smallness bound on $\|\Delta_1\|$. Under the condition $C_1(1+C_2)<1$ (with explicit constants $C_1$, $C_2$ depending on the spin set, boundary energy, and a metric $D$), there exists a unique limiting correlation function $\rho$ satisfying $\rho=\delta+\mathscr{K}\rho$, and finite-volume functions $\rho_\Lambda$ converge to $\rho$ as the volume grows. The results provide a universal Gibbs representation based on transition energies, and relate to vacuum potentials $\Phi$ by reducing the smallness condition to a bound on $\|\Phi\|$, thereby extending convergence and representation beyond sums of pairwise interactions. This work generalizes the correlation-function approach to broader statistical-mechanical systems and establishes a rigorous infinite-volume limit via contraction mappings.
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.