System of recursive equations for the partition functions of 1D models

TL;DR

This paper derives recursive equations for partition functions of 1D models with nearest-neighbor interactions and analyzes their solutions under certain parameter conditions.

Contribution

It introduces systems of recursive equations for various partition functions in 1D models with nearest-neighbor interactions and characterizes their solutions.

Findings

Derived recursive equations for partition functions.

Identified conditions for solutions of the recursive systems.

Provided analysis of solutions based on model parameters.

Abstract

In this note we consider several kind of partition functions of one-dimensional models with nearest - neighbor interactions $I_n, n\in \mathbf{Z}$ and spin values $\pm 1$. We derive systems of recursive equations for each kind of such functions. These systems depend on parameters $I_n, n\in \mathbf{Z}$. Under some conditions on the parameters we describe solutions of the systems of recursive equations.